Derivative

<h2 id="definition">Definition</h2>
<h3>As a limit</h3>
A <a href="/facts/Function_of_a_real_variable/bbNG3DnN">function of a real variable</a> 
 
 
 
 f
 (
 x
 )
 
 
 {\displaystyle f(x)}
 
 is <a href="/facts/Differentiable_function/jQqTgLk1">differentiable</a> at a point 
 
 
 
 a
 
 
 {\displaystyle a}
 
 of its <a href="/facts/Domain_of_a_function/FQvoeUpX">domain</a>, if its domain contains an <a href="/facts/Open_interval/e9se3vTe">open interval</a> containing ⁠
 
 
 
 a
 
 
 {\displaystyle a}
 
⁠, and the <a href="/facts/Limit_(mathematics)/j8JBhj4e">limit</a>

L
        =
        
          lim
          
            h
            →
            0
          
        
        
          
            
              f
              (
              a
              +
              h
              )
              −
              f
              (
              a
              )
            
            h
          
        
      
    
    {\displaystyle L=\lim _{h\to 0}{\frac {f(a+h)-f(a)}{h}}}

exists.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> This means that, for every positive <a href="/facts/Real_number/R02gw5Pb">real number</a> ⁠
 
 
 
 ε
 
 
 {\displaystyle \varepsilon }
 
⁠, there exists a positive real number 
 
 
 
 δ
 
 
 {\displaystyle \delta }
 
 such that, for every 
 
 
 
 h
 
 
 {\displaystyle h}
 
 such that 
 
 
 
 
 |
 
 h
 
 |
 
 <
 δ
 
 
 {\displaystyle |h|<\delta }
 
 and 
 
 
 
 h
 ≠
 0
 
 
 {\displaystyle h\neq 0}
 
 then 
 
 
 
 f
 (
 a
 +
 h
 )
 
 
 {\displaystyle f(a+h)}
 
 is defined, and

|
 
 L
 −
 
 
 
 f
 (
 a
 +
 h
 )
 −
 f
 (
 a
 )
 
 h
 
 
 
 |
 
 <
 ε
 ,
 
 
 {\displaystyle \left|L-{\frac {f(a+h)-f(a)}{h}}\right|<\varepsilon ,}

where the vertical bars denote the <a href="/facts/Absolute_value/dwJBpEs5">absolute value</a>. This is an example of the <a href="/facts/(%25CE%25B5%2c_%25CE%25B4)-definition_of_limit/3Yd7BdvM">(ε, δ)-definition of limit</a>.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>
If the function 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is differentiable at ⁠
 
 
 
 a
 
 
 {\displaystyle a}
 
⁠, that is if the limit 
 
 
 
 L
 
 
 {\displaystyle L}
 
 exists, then this limit is called the derivative of 
 
 
 
 f
 
 
 {\displaystyle f}
 
 at 
 
 
 
 a
 
 
 {\displaystyle a}
 
. Multiple notations for the derivative exist.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> The derivative of 
 
 
 
 f
 
 
 {\displaystyle f}
 
 at 
 
 
 
 a
 
 
 {\displaystyle a}
 
 can be denoted ⁠
 
 
 
 
 f
 ′
 
 (
 a
 )
 
 
 {\displaystyle f'(a)}
 
⁠, read as "⁠
 
 
 
 f
 
 
 {\displaystyle f}
 
⁠ prime of ⁠
 
 
 
 a
 
 
 {\displaystyle a}
 
⁠"; or it can be denoted ⁠
 
 
 
 
 
 
 
 d
 f
 
 
 d
 x
 
 
 
 (
 a
 )
 
 
 
 {\displaystyle \textstyle {\frac {df}{dx}}(a)}
 
⁠, read as "the derivative of 
 
 
 
 f
 
 
 {\displaystyle f}
 
 with respect to 
 
 
 
 x
 
 
 {\displaystyle x}
 
 at ⁠
 
 
 
 a
 
 
 {\displaystyle a}
 
⁠" or "⁠
 
 
 
 d
 f
 
 
 {\displaystyle df}
 
⁠ by (or over) 
 
 
 
 d
 x
 
 
 {\displaystyle dx}
 
 at ⁠
 
 
 
 a
 
 
 {\displaystyle a}
 
⁠". See § Notation below. If 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is a function that has a derivative at every point in its <a href="/facts/Domain_of_a_function/FQvoeUpX">domain</a>, then a function can be defined by mapping every point 
 
 
 
 x
 
 
 {\displaystyle x}
 
 to the value of the derivative of 
 
 
 
 f
 
 
 {\displaystyle f}
 
 at 
 
 
 
 x
 
 
 {\displaystyle x}
 
. This function is written 
 
 
 
 
 f
 ′
 
 
 
 {\displaystyle f'}
 
 and is called the derivative function or the derivative of ⁠
 
 
 
 f
 
 
 {\displaystyle f}
 
⁠. The function 
 
 
 
 f
 
 
 {\displaystyle f}
 
 sometimes has a derivative at most, but not all, points of its domain. The function whose value at 
 
 
 
 a
 
 
 {\displaystyle a}
 
 equals 
 
 
 
 
 f
 ′
 
 (
 a
 )
 
 
 {\displaystyle f'(a)}
 
 whenever 
 
 
 
 
 f
 ′
 
 (
 a
 )
 
 
 {\displaystyle f'(a)}
 
 is defined and elsewhere is undefined is also called the derivative of ⁠
 
 
 
 f
 
 
 {\displaystyle f}
 
⁠. It is still a function, but its domain may be smaller than the domain of 
 
 
 
 f
 
 
 {\displaystyle f}
 
.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>
For example, let 
 
 
 
 f
 
 
 {\displaystyle f}
 
 be the squaring function: 
 
 
 
 f
 (
 x
 )
 =
 
 x
 
 2
 
 
 
 
 {\displaystyle f(x)=x^{2}}
 
. Then the quotient in the definition of the derivative is<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>

f
              (
              a
              +
              h
              )
              −
              f
              (
              a
              )
            
            h
          
        
        =
        
          
            
              (
              a
              +
              h
              
                )
                
                  2
                
              
              −
              
                a
                
                  2
                
              
            
            h
          
        
        =
        
          
            
              
                a
                
                  2
                
              
              +
              2
              a
              h
              +
              
                h
                
                  2
                
              
              −
              
                a
                
                  2
                
              
            
            h
          
        
        =
        2
        a
        +
        h
        .
      
    
    {\displaystyle {\frac {f(a+h)-f(a)}{h}}={\frac {(a+h)^{2}-a^{2}}{h}}={\frac {a^{2}+2ah+h^{2}-a^{2}}{h}}=2a+h.}

The division in the last step is valid as long as 
 
 
 
 h
 ≠
 0
 
 
 {\displaystyle h\neq 0}
 
. The closer 
 
 
 
 h
 
 
 {\displaystyle h}
 
 is to ⁠
 
 
 
 0
 
 
 {\displaystyle 0}
 
⁠, the closer this expression becomes to the value 
 
 
 
 2
 a
 
 
 {\displaystyle 2a}
 
. The limit exists, and for every input 
 
 
 
 a
 
 
 {\displaystyle a}
 
 the limit is 
 
 
 
 2
 a
 
 
 {\displaystyle 2a}
 
. So, the derivative of the squaring function is the doubling function: ⁠
 
 
 
 
 f
 ′
 
 (
 x
 )
 =
 2
 x
 
 
 {\displaystyle f'(x)=2x}
 
⁠.
The ratio in the definition of the derivative is the slope of the line through two points on the graph of the function ⁠
 
 
 
 f
 
 
 {\displaystyle f}
 
⁠, specifically the points 
 
 
 
 (
 a
 ,
 f
 (
 a
 )
 )
 
 
 {\displaystyle (a,f(a))}
 
 and 
 
 
 
 (
 a
 +
 h
 ,
 f
 (
 a
 +
 h
 )
 )
 
 
 {\displaystyle (a+h,f(a+h))}
 
. As 
 
 
 
 h
 
 
 {\displaystyle h}
 
 is made smaller, these points grow closer together, and the slope of this line approaches the limiting value, the slope of the <a href="/facts/Tangent/sF0jH3EI">tangent</a> to the graph of 
 
 
 
 f
 
 
 {\displaystyle f}
 
 at 
 
 
 
 a
 
 
 {\displaystyle a}
 
. In other words, the derivative is the slope of the tangent.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>

<h3>Using infinitesimals</h3>
One way to think of the derivative 
 
 
 
 
 
 
 d
 f
 
 
 d
 x
 
 
 
 (
 a
 )
 
 
 {\textstyle {\frac {df}{dx}}(a)}
 
 is as the ratio of an <a href="/facts/Infinitesimal/JKcTMIgD">infinitesimal</a> change in the output of the function 
 
 
 
 f
 
 
 {\displaystyle f}
 
 to an infinitesimal change in its input.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> In order to make this intuition rigorous, a system of rules for manipulating infinitesimal quantities is required.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a> The system of <a href="/facts/Hyperreal_number/Kvf09qpo">hyperreal numbers</a> is a way of treating <a href="/facts/Infinity/cF506uD7">infinite</a> and infinitesimal quantities. The hyperreals are an <a href="/facts/Field_extension/L7yIDtyK">extension</a> of the <a href="/facts/Real_number/R02gw5Pb">real numbers</a> that contain numbers greater than anything of the form 
 
 
 
 1
 +
 1
 +
 ⋯
 +
 1
 
 
 {\displaystyle 1+1+\cdots +1}
 
 for any finite number of terms. Such numbers are infinite, and their <a href="/facts/Multiplicative_inverse/yczdYyCw">reciprocals</a> are infinitesimals. The application of hyperreal numbers to the foundations of calculus is called <a href="/facts/Nonstandard_analysis/lEpNpWMB">nonstandard analysis</a>. This provides a way to define the basic concepts of calculus such as the derivative and integral in terms of infinitesimals, thereby giving a precise meaning to the 
 
 
 
 d
 
 
 {\displaystyle d}
 
 in the Leibniz notation. Thus, the derivative of 
 
 
 
 f
 (
 x
 )
 
 
 {\displaystyle f(x)}
 
 becomes 
 
 
 
 
 f
 ′
 
 (
 x
 )
 =
 st
 ⁡
 
 (
 
 
 
 f
 (
 x
 +
 d
 x
 )
 −
 f
 (
 x
 )
 
 
 d
 x
 
 
 
 )
 
 
 
 {\displaystyle f'(x)=\operatorname {st} \left({\frac {f(x+dx)-f(x)}{dx}}\right)}
 
 for an arbitrary infinitesimal ⁠
 
 
 
 d
 x
 
 
 {\displaystyle dx}
 
⁠, where 
 
 
 
 st
 
 
 {\displaystyle \operatorname {st} }
 
 denotes the <a href="/facts/Standard_part_function/fTuL5IXj">standard part function</a>, which "rounds off" each finite hyperreal to the nearest real.<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a> Taking the squaring function 
 
 
 
 f
 (
 x
 )
 =
 
 x
 
 2
 
 
 
 
 {\displaystyle f(x)=x^{2}}
 
 as an example again,

f
                  ′
                
                (
                x
                )
              
              
                
                =
                st
                ⁡
                
                  (
                  
                    
                      
                        
                          x
                          
                            2
                          
                        
                        +
                        2
                        x
                        ⋅
                        d
                        x
                        +
                        (
                        d
                        x
                        
                          )
                          
                            2
                          
                        
                        −
                        
                          x
                          
                            2
                          
                        
                      
                      
                        d
                        x
                      
                    
                  
                  )
                
              
            
            
              
              
                
                =
                st
                ⁡
                
                  (
                  
                    
                      
                        2
                        x
                        ⋅
                        d
                        x
                        +
                        (
                        d
                        x
                        
                          )
                          
                            2
                          
                        
                      
                      
                        d
                        x
                      
                    
                  
                  )
                
              
            
            
              
              
                
                =
                st
                ⁡
                
                  (
                  
                    
                      
                        
                          2
                          x
                          ⋅
                          d
                          x
                        
                        
                          d
                          x
                        
                      
                    
                    +
                    
                      
                        
                          (
                          d
                          x
                          
                            )
                            
                              2
                            
                          
                        
                        
                          d
                          x
                        
                      
                    
                  
                  )
                
              
            
            
              
              
                
                =
                st
                ⁡
                
                  (
                  
                    2
                    x
                    +
                    d
                    x
                  
                  )
                
              
            
            
              
              
                
                =
                2
                x
                .
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}f'(x)&=\operatorname {st} \left({\frac {x^{2}+2x\cdot dx+(dx)^{2}-x^{2}}{dx}}\right)\\&=\operatorname {st} \left({\frac {2x\cdot dx+(dx)^{2}}{dx}}\right)\\&=\operatorname {st} \left({\frac {2x\cdot dx}{dx}}+{\frac {(dx)^{2}}{dx}}\right)\\&=\operatorname {st} \left(2x+dx\right)\\&=2x.\end{aligned}}}

<h2 id="continuity-and-differentiability">Continuity and differentiability</h2>

If 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is <a href="/facts/Differentiable/jQqTgLk1">differentiable</a> at ⁠
 
 
 
 a
 
 
 {\displaystyle a}
 
⁠, then 
 
 
 
 f
 
 
 {\displaystyle f}
 
 must also be <a href="/facts/Continuous_function/sKbl02pB">continuous</a> at 
 
 
 
 a
 
 
 {\displaystyle a}
 
.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a> As an example, choose a point 
 
 
 
 a
 
 
 {\displaystyle a}
 
 and let 
 
 
 
 f
 
 
 {\displaystyle f}
 
 be the <a href="/facts/Step_function/VnrEjo4B">step function</a> that returns the value 1 for all 
 
 
 
 x
 
 
 {\displaystyle x}
 
 less than ⁠
 
 
 
 a
 
 
 {\displaystyle a}
 
⁠, and returns a different value 10 for all 
 
 
 
 x
 
 
 {\displaystyle x}
 
 greater than or equal to 
 
 
 
 a
 
 
 {\displaystyle a}
 
. The function 
 
 
 
 f
 
 
 {\displaystyle f}
 
 cannot have a derivative at 
 
 
 
 a
 
 
 {\displaystyle a}
 
. If 
 
 
 
 h
 
 
 {\displaystyle h}
 
 is negative, then 
 
 
 
 a
 +
 h
 
 
 {\displaystyle a+h}
 
 is on the low part of the step, so the secant line from 
 
 
 
 a
 
 
 {\displaystyle a}
 
 to 
 
 
 
 a
 +
 h
 
 
 {\displaystyle a+h}
 
 is very steep; as 
 
 
 
 h
 
 
 {\displaystyle h}
 
 tends to zero, the slope tends to infinity. If 
 
 
 
 h
 
 
 {\displaystyle h}
 
 is positive, then 
 
 
 
 a
 +
 h
 
 
 {\displaystyle a+h}
 
 is on the high part of the step, so the secant line from 
 
 
 
 a
 
 
 {\displaystyle a}
 
 to 
 
 
 
 a
 +
 h
 
 
 {\displaystyle a+h}
 
 has slope zero. Consequently, the secant lines do not approach any single slope, so the limit of the difference quotient does not exist. However, even if a function is continuous at a point, it may not be differentiable there. For example, the <a href="/facts/Absolute_value/dwJBpEs5">absolute value</a> function given by 
 
 
 
 f
 (
 x
 )
 =
 
 |
 
 x
 
 |
 
 
 
 {\displaystyle f(x)=|x|}
 
 is continuous at ⁠
 
 
 
 x
 =
 0
 
 
 {\displaystyle x=0}
 
⁠, but it is not differentiable there. If 
 
 
 
 h
 
 
 {\displaystyle h}
 
 is positive, then the slope of the secant line from 0 to 
 
 
 
 h
 
 
 {\displaystyle h}
 
 is one; if 
 
 
 
 h
 
 
 {\displaystyle h}
 
 is negative, then the slope of the secant line from 
 
 
 
 0
 
 
 {\displaystyle 0}
 
 to 
 
 
 
 h
 
 
 {\displaystyle h}
 
 is ⁠
 
 
 
 −
 1
 
 
 {\displaystyle -1}
 
⁠.<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a> This can be seen graphically as a "kink" or a "cusp" in the graph at 
 
 
 
 x
 =
 0
 
 
 {\displaystyle x=0}
 
. Even a function with a smooth graph is not differentiable at a point where its <a href="/facts/Vertical_tangent/GC5NTJEE">tangent is vertical</a>: For instance, the function given by 
 
 
 
 f
 (
 x
 )
 =
 
 x
 
 1
 
 /
 
 3
 
 
 
 
 {\displaystyle f(x)=x^{1/3}}
 
 is not differentiable at 
 
 
 
 x
 =
 0
 
 
 {\displaystyle x=0}
 
. In summary, a function that has a derivative is continuous, but there are continuous functions that do not have a derivative.<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a>
Most functions that occur in practice have derivatives at all points or <a href="/facts/Almost_everywhere/1quRFtJE">almost every</a> point. Early in the <a href="/facts/History_of_calculus/kJbq5owp">history of calculus</a>, many mathematicians assumed that a continuous function was differentiable at most points.<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a> Under mild conditions (for example, if the function is a <a href="/facts/Monotone_function/MjQK0YL2">monotone</a> or a <a href="/facts/Lipschitz_function/Hw20EPEU">Lipschitz function</a>), this is true. However, in 1872, Weierstrass found the first example of a function that is continuous everywhere but differentiable nowhere. This example is now known as the <a href="/facts/Weierstrass_function/m9jU4LE5">Weierstrass function</a>.<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a> In 1931, <a href="/facts/Stefan_Banach/b1qHSNg2">Stefan Banach</a> proved that the set of functions that have a derivative at some point is a <a href="/facts/Meager_set/AxTk8frw">meager set</a> in the space of all continuous functions. Informally, this means that hardly any random continuous functions have a derivative at even one point.<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a>

<h2 id="notation">Notation</h2>
Main article: <a href="/facts/Notation_for_differentiation/PTCbwS0Y">Notation for differentiation</a>
One common way of writing the derivative of a function is <a href="/facts/Leibniz_notation/FLytWNf9">Leibniz notation</a>, introduced by <a href="/facts/Gottfried_Leibniz/cTc2er83">Gottfried Wilhelm Leibniz</a> in 1675, which denotes a derivative as the quotient of two <a href="/facts/Differential_(mathematics)/YmSDWCol">differentials</a>, such as 
 
 
 
 d
 y
 
 
 {\displaystyle dy}
 
 and ⁠
 
 
 
 d
 x
 
 
 {\displaystyle dx}
 
⁠.<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a> It is still commonly used when the equation 
 
 
 
 y
 =
 f
 (
 x
 )
 
 
 {\displaystyle y=f(x)}
 
 is viewed as a functional relationship between <a href="/facts/Dependent_and_independent_variables/dINfzIF2">dependent and independent variables</a>. The first derivative is denoted by ⁠
 
 
 
 
 
 
 
 d
 y
 
 
 d
 x
 
 
 
 
 
 
 {\displaystyle \textstyle {\frac {dy}{dx}}}
 
⁠, read as "the derivative of 
 
 
 
 y
 
 
 {\displaystyle y}
 
 with respect to ⁠
 
 
 
 x
 
 
 {\displaystyle x}
 
⁠".<a class="footnote-ref" id="fnref:18" href="#fn:18">18</a> This derivative can alternately be treated as the application of a <a href="/facts/Differential_operator/Nr15J0IE">differential operator</a> to a function, 
 
 
 
 
 
 
 d
 y
 
 
 d
 x
 
 
 
 =
 
 
 d
 
 d
 x
 
 
 
 f
 (
 x
 )
 .
 
 
 {\textstyle {\frac {dy}{dx}}={\frac {d}{dx}}f(x).}
 
 Higher derivatives are expressed using the notation 
 
 
 
 
 
 
 
 d
 
 n
 
 
 y
 
 
 d
 
 x
 
 n
 
 
 
 
 
 
 
 {\textstyle {\frac {d^{n}y}{dx^{n}}}}
 
 for the 
 
 
 
 n
 
 
 {\displaystyle n}
 
-th derivative of 
 
 
 
 y
 =
 f
 (
 x
 )
 
 
 {\displaystyle y=f(x)}
 
. These are abbreviations for multiple applications of the derivative operator; for example, 
 
 
 
 
 
 
 
 d
 
 2
 
 
 y
 
 
 d
 
 x
 
 2
 
 
 
 
 
 =
 
 
 d
 
 d
 x
 
 
 
 
 
 (
 
 
 
 
 d
 
 d
 x
 
 
 
 f
 (
 x
 )
 
 
 )
 
 
 .
 
 
 {\textstyle {\frac {d^{2}y}{dx^{2}}}={\frac {d}{dx}}{\Bigl (}{\frac {d}{dx}}f(x){\Bigr )}.}
 
<a class="footnote-ref" id="fnref:19" href="#fn:19">19</a> Unlike some alternatives, Leibniz notation involves explicit specification of the variable for differentiation, in the denominator, which removes ambiguity when working with multiple interrelated quantities. The derivative of a <a href="/facts/Function_composition/r5Pvnmth">composed function</a> can be expressed using the <a href="/facts/Chain_rule/aMLYxP0x">chain rule</a>: if 
 
 
 
 u
 =
 g
 (
 x
 )
 
 
 {\displaystyle u=g(x)}
 
 and 
 
 
 
 y
 =
 f
 (
 g
 (
 x
 )
 )
 
 
 {\displaystyle y=f(g(x))}
 
 then 
 
 
 
 
 
 
 d
 y
 
 
 d
 x
 
 
 
 =
 
 
 
 d
 y
 
 
 d
 u
 
 
 
 ⋅
 
 
 
 d
 u
 
 
 d
 x
 
 
 
 .
 
 
 {\textstyle {\frac {dy}{dx}}={\frac {dy}{du}}\cdot {\frac {du}{dx}}.}
 
<a class="footnote-ref" id="fnref:20" href="#fn:20">20</a>
Another common notation for differentiation is by using the <a href="/facts/Prime_(symbol)/3p9EvU5X">prime mark</a> in the symbol of a function ⁠
 
 
 
 f
 (
 x
 )
 
 
 {\displaystyle f(x)}
 
⁠. This notation, due to <a href="/facts/Joseph-Louis_Lagrange/bTQUaSTv">Joseph-Louis Lagrange</a>, is now known as prime notation.<a class="footnote-ref" id="fnref:21" href="#fn:21">21</a> The first derivative is written as ⁠
 
 
 
 
 f
 ′
 
 (
 x
 )
 
 
 {\displaystyle f'(x)}
 
⁠, read as "⁠
 
 
 
 f
 
 
 {\displaystyle f}
 
⁠ prime of ⁠
 
 
 
 x
 
 
 {\displaystyle x}
 
⁠, or ⁠
 
 
 
 
 y
 ′
 
 
 
 {\displaystyle y'}
 
⁠, read as "⁠
 
 
 
 y
 
 
 {\displaystyle y}
 
⁠ prime".<a class="footnote-ref" id="fnref:22" href="#fn:22">22</a> Similarly, the second and the third derivatives can be written as 
 
 
 
 
 f
 ″
 
 
 
 {\displaystyle f''}
 
 and ⁠
 
 
 
 
 f
 ‴
 
 
 
 {\displaystyle f'''}
 
⁠, respectively.<a class="footnote-ref" id="fnref:23" href="#fn:23">23</a> For denoting the number of higher derivatives beyond this point, some authors use Roman numerals in <a href="/facts/Subscript_and_superscript/6tExBnkw">superscript</a>, whereas others place the number in parentheses, such as 
 
 
 
 
 f
 
 
 i
 v
 
 
 
 
 
 {\displaystyle f^{\mathrm {iv} }}
 
 or ⁠
 
 
 
 
 f
 
 (
 4
 )
 
 
 
 
 {\displaystyle f^{(4)}}
 
⁠.<a class="footnote-ref" id="fnref:24" href="#fn:24">24</a> The latter notation generalizes to yield the notation 
 
 
 
 
 f
 
 (
 n
 )
 
 
 
 
 {\displaystyle f^{(n)}}
 
 for the ⁠
 
 
 
 n
 
 
 {\displaystyle n}
 
⁠th derivative of ⁠
 
 
 
 f
 
 
 {\displaystyle f}
 
⁠.<a class="footnote-ref" id="fnref:25" href="#fn:25">25</a>
In <a href="/facts/Newton%2527s_notation/PTCbwS0Y">Newton's notation</a> or the dot notation, a dot is placed over a symbol to represent a time derivative. If 
 
 
 
 y
 
 
 {\displaystyle y}
 
 is a function of ⁠
 
 
 
 t
 
 
 {\displaystyle t}
 
⁠, then the first and second derivatives can be written as 
 
 
 
 
 
 
 y
 ˙
 
 
 
 
 
 {\displaystyle {\dot {y}}}
 
 and ⁠
 
 
 
 
 
 
 y
 ¨
 
 
 
 
 
 {\displaystyle {\ddot {y}}}
 
⁠, respectively. This notation is used exclusively for derivatives with respect to time or <a href="/facts/Arc_length/QgTrmLxY">arc length</a>. It is typically used in <a href="/facts/Differential_equation/9MGbE3Ca">differential equations</a> in <a href="/facts/Physics/a7aH2y08">physics</a> and <a href="/facts/Differential_geometry/c04XmyLu">differential geometry</a>.<a class="footnote-ref" id="fnref:26" href="#fn:26">26</a> However, the dot notation becomes unmanageable for high-order derivatives (of order 4 or more) and cannot deal with multiple independent variables.
Another notation is D-notation, which represents the differential operator by the symbol ⁠
 
 
 
 D
 
 
 {\displaystyle D}
 
⁠.<a class="footnote-ref" id="fnref:27" href="#fn:27">27</a> The first derivative is written 
 
 
 
 D
 f
 (
 x
 )
 
 
 {\displaystyle Df(x)}
 
 and higher derivatives are written with a superscript, so the 
 
 
 
 n
 
 
 {\displaystyle n}
 
-th derivative is ⁠
 
 
 
 
 D
 
 n
 
 
 f
 (
 x
 )
 
 
 {\displaystyle D^{n}f(x)}
 
⁠. This notation is sometimes called Euler notation, although it seems that <a href="/facts/Leonhard_Euler/z2fsbjgj">Leonhard Euler</a> did not use it, and the notation was introduced by <a href="/facts/Louis_Fran%25C3%25A7ois_Antoine_Arbogast/hBfMpb0b">Louis François Antoine Arbogast</a>.<a class="footnote-ref" id="fnref:28" href="#fn:28">28</a> To indicate a partial derivative, the variable differentiated by is indicated with a subscript, for example given the function ⁠
 
 
 
 u
 =
 f
 (
 x
 ,
 y
 )
 
 
 {\displaystyle u=f(x,y)}
 
⁠, its partial derivative with respect to 
 
 
 
 x
 
 
 {\displaystyle x}
 
 can be written 
 
 
 
 
 D
 
 x
 
 
 u
 
 
 {\displaystyle D_{x}u}
 
 or ⁠
 
 
 
 
 D
 
 x
 
 
 f
 (
 x
 ,
 y
 )
 
 
 {\displaystyle D_{x}f(x,y)}
 
⁠. Higher partial derivatives can be indicated by superscripts or multiple subscripts, e.g. 
 
 
 
 
 D
 
 x
 y
 
 
 f
 (
 x
 ,
 y
 )
 =
 
 
 ∂
 
 ∂
 y
 
 
 
 
 
 (
 
 
 
 
 ∂
 
 ∂
 x
 
 
 
 f
 (
 x
 ,
 y
 )
 
 
 )
 
 
 
 
 {\textstyle D_{xy}f(x,y)={\frac {\partial }{\partial y}}{\Bigl (}{\frac {\partial }{\partial x}}f(x,y){\Bigr )}}
 
 and ⁠
 
 
 
 
 
 D
 
 x
 
 
 2
 
 
 f
 (
 x
 ,
 y
 )
 =
 
 
 ∂
 
 ∂
 x
 
 
 
 
 
 (
 
 
 
 
 ∂
 
 ∂
 x
 
 
 
 f
 (
 x
 ,
 y
 )
 
 
 )
 
 
 
 
 
 {\displaystyle \textstyle D_{x}^{2}f(x,y)={\frac {\partial }{\partial x}}{\Bigl (}{\frac {\partial }{\partial x}}f(x,y){\Bigr )}}
 
⁠.<a class="footnote-ref" id="fnref:29" href="#fn:29">29</a>

<h2 id="rules-of-computation">Rules of computation</h2>
Main article: <a href="/facts/Differentiation_rules/9q1b7Ev6">Differentiation rules</a>
In principle, the derivative of a function can be computed from the definition by considering the difference quotient and computing its limit. Once the derivatives of a few simple functions are known, the derivatives of other functions are more easily computed using rules for obtaining derivatives of more complicated functions from simpler ones. This process of finding a derivative is known as differentiation.<a class="footnote-ref" id="fnref:30" href="#fn:30">30</a>

<h3>Rules for basic functions</h3>
The following are the rules for the derivatives of the most common basic functions. Here, 
 
 
 
 a
 
 
 {\displaystyle a}
 
 is a real number, and 
 
 
 
 e
 
 
 {\displaystyle e}
 
 is <a href="/facts/E_(mathematical_constant)/coEtSBk8">the base of the natural logarithm, approximately 2.71828</a>.<a class="footnote-ref" id="fnref:31" href="#fn:31">31</a> 

<ul><li><a href="/facts/Power_rule/V6wJxLlp">Derivatives of powers</a>:

d
 
 d
 x
 
 
 
 
 x
 
 a
 
 
 =
 a
 
 x
 
 a
 −
 1
 
 
 
 
 {\displaystyle {\frac {d}{dx}}x^{a}=ax^{a-1}}
 
</li>
<li>Functions of <a href="/facts/Exponential_function/ewlYxvR2">exponential</a>, <a href="/facts/Natural_logarithm/PnreKEzI">natural logarithm</a>, and <a href="/facts/Logarithm/QeXaV97q">logarithm</a> with general base:

d
            
              d
              x
            
          
        
        
          e
          
            x
          
        
        =
        
          e
          
            x
          
        
      
    
    {\displaystyle {\frac {d}{dx}}e^{x}=e^{x}}

d
            
              d
              x
            
          
        
        
          a
          
            x
          
        
        =
        
          a
          
            x
          
        
        ln
        ⁡
        (
        a
        )
      
    
    {\displaystyle {\frac {d}{dx}}a^{x}=a^{x}\ln(a)}
  
, for 
  
    
      
        a
        >
        0
      
    
    {\displaystyle a>0}

d
            
              d
              x
            
          
        
        ln
        ⁡
        (
        x
        )
        =
        
          
            1
            x
          
        
      
    
    {\displaystyle {\frac {d}{dx}}\ln(x)={\frac {1}{x}}}
  
, for 
  
    
      
        x
        >
        0
      
    
    {\displaystyle x>0}

d
 
 d
 x
 
 
 
 
 log
 
 a
 
 
 ⁡
 (
 x
 )
 =
 
 
 1
 
 x
 ln
 ⁡
 (
 a
 )
 
 
 
 
 
 {\displaystyle {\frac {d}{dx}}\log _{a}(x)={\frac {1}{x\ln(a)}}}
 
, for 
 
 
 
 x
 ,
 a
 >
 0
 
 
 {\displaystyle x,a>0}
 
</li>
<li><a href="/facts/Trigonometric_functions/czej7ur0">Trigonometric functions</a>:

d
            
              d
              x
            
          
        
        sin
        ⁡
        (
        x
        )
        =
        cos
        ⁡
        (
        x
        )
      
    
    {\displaystyle {\frac {d}{dx}}\sin(x)=\cos(x)}

d
            
              d
              x
            
          
        
        cos
        ⁡
        (
        x
        )
        =
        −
        sin
        ⁡
        (
        x
        )
      
    
    {\displaystyle {\frac {d}{dx}}\cos(x)=-\sin(x)}

d
 
 d
 x
 
 
 
 tan
 ⁡
 (
 x
 )
 =
 
 sec
 
 2
 
 
 ⁡
 (
 x
 )
 =
 
 
 1
 
 
 cos
 
 2
 
 
 ⁡
 (
 x
 )
 
 
 
 =
 1
 +
 
 tan
 
 2
 
 
 ⁡
 (
 x
 )
 
 
 {\displaystyle {\frac {d}{dx}}\tan(x)=\sec ^{2}(x)={\frac {1}{\cos ^{2}(x)}}=1+\tan ^{2}(x)}
 
</li>
<li><a href="/facts/Inverse_trigonometric_functions/bVwU9FPG">Inverse trigonometric functions</a>:

d
 
 d
 x
 
 
 
 arcsin
 ⁡
 (
 x
 )
 =
 
 
 1
 
 1
 −
 
 x
 
 2
 
 
 
 
 
 
 
 {\displaystyle {\frac {d}{dx}}\arcsin(x)={\frac {1}{\sqrt {1-x^{2}}}}}
 
, for 
 
 
 
 −
 1
 <
 x
 <
 1
 
 
 {\displaystyle -1<x<1}

d
 
 d
 x
 
 
 
 arccos
 ⁡
 (
 x
 )
 =
 −
 
 
 1
 
 1
 −
 
 x
 
 2
 
 
 
 
 
 
 
 {\displaystyle {\frac {d}{dx}}\arccos(x)=-{\frac {1}{\sqrt {1-x^{2}}}}}
 
, for 
 
 
 
 −
 1
 <
 x
 <
 1
 
 
 {\displaystyle -1<x<1}

d
 
 d
 x
 
 
 
 arctan
 ⁡
 (
 x
 )
 =
 
 
 1
 
 1
 +
 
 x
 
 2
 
 
 
 
 
 
 
 {\displaystyle {\frac {d}{dx}}\arctan(x)={\frac {1}{1+x^{2}}}}
 
</li></ul>
<h3>Rules for combined functions </h3>
Given that the 
 
 
 
 f
 
 
 {\displaystyle f}
 
 and 
 
 
 
 g
 
 
 {\displaystyle g}
 
 are the functions. The following are some of the most basic rules for deducing the derivative of functions from derivatives of basic functions.<a class="footnote-ref" id="fnref:32" href="#fn:32">32</a>

<ul><li>Constant rule: if 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is constant, then for all ⁠
 
 
 
 x
 
 
 {\displaystyle x}
 
⁠,

f
 ′
 
 (
 x
 )
 =
 0.
 
 
 {\displaystyle f'(x)=0.}
 
</li>
<li><a href="/facts/Linearity_of_differentiation/zWJ2Pqap">Sum rule</a>:

(
 α
 f
 +
 β
 g
 
 )
 ′
 
 =
 α
 
 f
 ′
 
 +
 β
 
 g
 ′
 
 
 
 {\displaystyle (\alpha f+\beta g)'=\alpha f'+\beta g'}
 
 for all functions 
 
 
 
 f
 
 
 {\displaystyle f}
 
 and 
 
 
 
 g
 
 
 {\displaystyle g}
 
 and all real numbers 
 
 
 
 α
 
 
 {\displaystyle \alpha }
 
 and ⁠
 
 
 
 β
 
 
 {\displaystyle \beta }
 
⁠.</li>
<li><a href="/facts/Product_rule/MyKxsP5t">Product rule</a>:

(
 f
 g
 
 )
 ′
 
 =
 
 f
 ′
 
 g
 +
 f
 
 g
 ′
 
 
 
 {\displaystyle (fg)'=f'g+fg'}
 
 for all functions 
 
 
 
 f
 
 
 {\displaystyle f}
 
 and ⁠
 
 
 
 g
 
 
 {\displaystyle g}
 
⁠. As a special case, this rule includes the fact 
 
 
 
 (
 α
 f
 
 )
 ′
 
 =
 α
 
 f
 ′
 
 
 
 {\displaystyle (\alpha f)'=\alpha f'}
 
 whenever 
 
 
 
 α
 
 
 {\displaystyle \alpha }
 
 is a constant because 
 
 
 
 
 α
 ′
 
 f
 =
 0
 ⋅
 f
 =
 0
 
 
 {\displaystyle \alpha 'f=0\cdot f=0}
 
 by the constant rule.</li>
<li><a href="/facts/Quotient_rule/GBvTloJ5">Quotient rule</a>:

(
 
 
 f
 g
 
 
 )
 
 ′
 
 =
 
 
 
 
 f
 ′
 
 g
 −
 f
 
 g
 ′
 
 
 
 g
 
 2
 
 
 
 
 
 
 {\displaystyle \left({\frac {f}{g}}\right)'={\frac {f'g-fg'}{g^{2}}}}
 
 for all functions 
 
 
 
 f
 
 
 {\displaystyle f}
 
 and 
 
 
 
 g
 
 
 {\displaystyle g}
 
 at all inputs where g ≠ 0.</li>
<li><a href="/facts/Chain_rule/aMLYxP0x">Chain rule</a> for <a href="/facts/Function_composition/r5Pvnmth">composite functions</a>: If ⁠
 
 
 
 f
 (
 x
 )
 =
 h
 (
 g
 (
 x
 )
 )
 
 
 {\displaystyle f(x)=h(g(x))}
 
⁠, then

f
 ′
 
 (
 x
 )
 =
 
 h
 ′
 
 (
 g
 (
 x
 )
 )
 ⋅
 
 g
 ′
 
 (
 x
 )
 .
 
 
 {\displaystyle f'(x)=h'(g(x))\cdot g'(x).}
 
</li></ul>
<h3>Computation example</h3>
The derivative of the function given by 
 
 
 
 f
 (
 x
 )
 =
 
 x
 
 4
 
 
 +
 sin
 ⁡
 
 (
 
 x
 
 2
 
 
 )
 
 −
 ln
 ⁡
 (
 x
 )
 
 e
 
 x
 
 
 +
 7
 
 
 {\displaystyle f(x)=x^{4}+\sin \left(x^{2}\right)-\ln(x)e^{x}+7}
 
 is

f
                  ′
                
                (
                x
                )
              
              
                
                =
                4
                
                  x
                  
                    (
                    4
                    −
                    1
                    )
                  
                
                +
                
                  
                    
                      d
                      
                        (
                        
                          x
                          
                            2
                          
                        
                        )
                      
                    
                    
                      d
                      x
                    
                  
                
                cos
                ⁡
                
                  (
                  
                    x
                    
                      2
                    
                  
                  )
                
                −
                
                  
                    
                      d
                      
                        (
                        
                          ln
                          ⁡
                          
                            x
                          
                        
                        )
                      
                    
                    
                      d
                      x
                    
                  
                
                
                  e
                  
                    x
                  
                
                −
                ln
                ⁡
                (
                x
                )
                
                  
                    
                      d
                      
                        (
                        
                          e
                          
                            x
                          
                        
                        )
                      
                    
                    
                      d
                      x
                    
                  
                
                +
                0
              
            
            
              
              
                
                =
                4
                
                  x
                  
                    3
                  
                
                +
                2
                x
                cos
                ⁡
                
                  (
                  
                    x
                    
                      2
                    
                  
                  )
                
                −
                
                  
                    1
                    x
                  
                
                
                  e
                  
                    x
                  
                
                −
                ln
                ⁡
                (
                x
                )
                
                  e
                  
                    x
                  
                
                .
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}f'(x)&=4x^{(4-1)}+{\frac {d\left(x^{2}\right)}{dx}}\cos \left(x^{2}\right)-{\frac {d\left(\ln {x}\right)}{dx}}e^{x}-\ln(x){\frac {d\left(e^{x}\right)}{dx}}+0\\&=4x^{3}+2x\cos \left(x^{2}\right)-{\frac {1}{x}}e^{x}-\ln(x)e^{x}.\end{aligned}}}

Here the second term was computed using the <a href="/facts/Chain_rule/aMLYxP0x">chain rule</a> and the third term using the <a href="/facts/Product_rule/MyKxsP5t">product rule</a>. The known derivatives of the elementary functions 
 
 
 
 
 x
 
 2
 
 
 
 
 {\displaystyle x^{2}}
 
, 
 
 
 
 
 x
 
 4
 
 
 
 
 {\displaystyle x^{4}}
 
, 
 
 
 
 sin
 ⁡
 (
 x
 )
 
 
 {\displaystyle \sin(x)}
 
, 
 
 
 
 ln
 ⁡
 (
 x
 )
 
 
 {\displaystyle \ln(x)}
 
, and 
 
 
 
 exp
 ⁡
 (
 x
 )
 =
 
 e
 
 x
 
 
 
 
 {\displaystyle \exp(x)=e^{x}}
 
, as well as the constant 
 
 
 
 7
 
 
 {\displaystyle 7}
 
, were also used.

<h2 id="higher-order-derivatives">Higher-order derivatives </h2>
Higher order derivatives are the result of differentiating a function repeatedly. Given that 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is a differentiable function, the derivative of 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is the first derivative, denoted as ⁠
 
 
 
 
 f
 ′
 
 
 
 {\displaystyle f'}
 
⁠. The derivative of 
 
 
 
 
 f
 ′
 
 
 
 {\displaystyle f'}
 
 is the <a href="/facts/Second_derivative/3aA6j4EY">second derivative</a>, denoted as ⁠
 
 
 
 
 f
 ″
 
 
 
 {\displaystyle f''}
 
⁠, and the derivative of 
 
 
 
 
 f
 ″
 
 
 
 {\displaystyle f''}
 
 is the <a href="/facts/Third_derivative/UPDze7CX">third derivative</a>, denoted as ⁠
 
 
 
 
 f
 ‴
 
 
 
 {\displaystyle f'''}
 
⁠. By continuing this process, if it exists, the ⁠
 
 
 
 n
 
 
 {\displaystyle n}
 
⁠th derivative is the derivative of the ⁠
 
 
 
 (
 n
 −
 1
 )
 
 
 {\displaystyle (n-1)}
 
⁠th derivative or the derivative of order ⁠
 
 
 
 n
 
 
 {\displaystyle n}
 
⁠. As has been discussed above, the generalization of derivative of a function 
 
 
 
 f
 
 
 {\displaystyle f}
 
 may be denoted as ⁠
 
 
 
 
 f
 
 (
 n
 )
 
 
 
 
 {\displaystyle f^{(n)}}
 
⁠.<a class="footnote-ref" id="fnref:33" href="#fn:33">33</a> A function that has 
 
 
 
 k
 
 
 {\displaystyle k}
 
 successive derivatives is called 
 
 
 
 k
 
 
 {\displaystyle k}
 
 times differentiable. If the 
 
 
 
 k
 
 
 {\displaystyle k}
 
-th derivative is continuous, then the function is said to be of <a href="/facts/Differentiability_class/mffL4ch2">differentiability class</a> ⁠
 
 
 
 
 C
 
 k
 
 
 
 
 {\displaystyle C^{k}}
 
⁠.<a class="footnote-ref" id="fnref:34" href="#fn:34">34</a> A function that has infinitely many derivatives is called infinitely differentiable or <a href="/facts/Smoothness/mffL4ch2">smooth</a>.<a class="footnote-ref" id="fnref:35" href="#fn:35">35</a> Any <a href="/facts/Polynomial/Lzak8VVx">polynomial</a> function is infinitely differentiable; taking derivatives repeatedly will eventually result in a <a href="/facts/Constant_function/BmPx6dn7">constant function</a>, and all subsequent derivatives of that function are zero.<a class="footnote-ref" id="fnref:36" href="#fn:36">36</a>
One <a href="/facts/Differential_calculus/uU1AAc4N">application of higher-order derivatives</a> is in <a href="/facts/Physics/a7aH2y08">physics</a>. Suppose that a function represents the position of an object at the time. The first derivative of that function is the <a href="/facts/Velocity/Q3o1LVDe">velocity</a> of an object with respect to time, the second derivative of the function is the <a href="/facts/Acceleration/UVDmjYvR">acceleration</a> of an object with respect to time,<a class="footnote-ref" id="fnref:37" href="#fn:37">37</a> and the third derivative is the <a href="/facts/Jerk_(physics)/8T5H2g0C">jerk</a>.<a class="footnote-ref" id="fnref:38" href="#fn:38">38</a>

<h2 id="in-other-dimensions">In other dimensions</h2>
See also: <a href="/facts/Vector_calculus/mXKIJLJE">Vector calculus</a> and <a href="/facts/Multivariable_calculus/PEs5SgUF">Multivariable calculus</a>
<h3>Vector-valued functions</h3>
A <a href="/facts/Vector-valued_function/qxu8aapt">vector-valued function</a> 
 
 
 
 
 y
 
 
 
 {\displaystyle \mathbf {y} }
 
 of a real variable sends real numbers to vectors in some <a href="/facts/Vector_space/M1pxMLx2">vector space</a> 
 
 
 
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {R} ^{n}}
 
. A vector-valued function can be split up into its coordinate functions 
 
 
 
 
 y
 
 1
 
 
 (
 t
 )
 ,
 
 y
 
 2
 
 
 (
 t
 )
 ,
 …
 ,
 
 y
 
 n
 
 
 (
 t
 )
 
 
 {\displaystyle y_{1}(t),y_{2}(t),\dots ,y_{n}(t)}
 
, meaning that 
 
 
 
 
 y
 
 =
 (
 
 y
 
 1
 
 
 (
 t
 )
 ,
 
 y
 
 2
 
 
 (
 t
 )
 ,
 …
 ,
 
 y
 
 n
 
 
 (
 t
 )
 )
 
 
 {\displaystyle \mathbf {y} =(y_{1}(t),y_{2}(t),\dots ,y_{n}(t))}
 
. This includes, for example, <a href="/facts/Parametric_curve/5Uvrlgqf">parametric curves</a> in 
 
 
 
 
 
 R
 
 
 2
 
 
 
 
 {\displaystyle \mathbb {R} ^{2}}
 
 or 
 
 
 
 
 
 R
 
 
 3
 
 
 
 
 {\displaystyle \mathbb {R} ^{3}}
 
. The coordinate functions are real-valued functions, so the above definition of derivative applies to them. The derivative of 
 
 
 
 
 y
 
 (
 t
 )
 
 
 {\displaystyle \mathbf {y} (t)}
 
 is defined to be the <a href="/facts/Vector_(geometric)/bCLrdksy">vector</a>, called the <a href="/facts/Differential_geometry_of_curves/UbbkBrs0">tangent vector</a>, whose coordinates are the derivatives of the coordinate functions. That is,<a class="footnote-ref" id="fnref:39" href="#fn:39">39</a>

y
          
          ′
        
        (
        t
        )
        =
        
          lim
          
            h
            →
            0
          
        
        
          
            
              
                y
              
              (
              t
              +
              h
              )
              −
              
                y
              
              (
              t
              )
            
            h
          
        
        ,
      
    
    {\displaystyle \mathbf {y} '(t)=\lim _{h\to 0}{\frac {\mathbf {y} (t+h)-\mathbf {y} (t)}{h}},}

if the limit exists. The subtraction in the numerator is the subtraction of vectors, not scalars. If the derivative of 
 
 
 
 
 y
 
 
 
 {\displaystyle \mathbf {y} }
 
 exists for every value of ⁠
 
 
 
 t
 
 
 {\displaystyle t}
 
⁠, then 
 
 
 
 
 
 y
 
 ′
 
 
 
 {\displaystyle \mathbf {y} '}
 
 is another vector-valued function.<a class="footnote-ref" id="fnref:40" href="#fn:40">40</a>

<h3>Partial derivatives</h3>
Main article: <a href="/facts/Partial_derivative/JWHsBkAu">Partial derivative</a>
Functions can depend upon <a href="/facts/Function_(mathematics)/CdReEKCh">more than one variable</a>. A <a href="/facts/Partial_derivative/JWHsBkAu">partial derivative</a> of a function of several variables is its derivative with respect to one of those variables, with the others held constant. Partial derivatives are used in <a href="/facts/Vector_calculus/mXKIJLJE">vector calculus</a> and <a href="/facts/Differential_geometry/c04XmyLu">differential geometry</a>. As with ordinary derivatives, multiple notations exist: the partial derivative of a function 
 
 
 
 f
 (
 x
 ,
 y
 ,
 …
 )
 
 
 {\displaystyle f(x,y,\dots )}
 
 with respect to the variable 
 
 
 
 x
 
 
 {\displaystyle x}
 
 is variously denoted by

f
 
 x
 
 
 
 
 {\displaystyle f_{x}}
 
, 
 
 
 
 
 f
 
 x
 
 ′
 
 
 
 {\displaystyle f'_{x}}
 
, 
 
 
 
 
 ∂
 
 x
 
 
 f
 
 
 {\displaystyle \partial _{x}f}
 
, 
 
 
 
 
 
 ∂
 
 ∂
 x
 
 
 
 f
 
 
 {\displaystyle {\frac {\partial }{\partial x}}f}
 
, or 
 
 
 
 
 
 
 ∂
 f
 
 
 ∂
 x
 
 
 
 
 
 {\displaystyle {\frac {\partial f}{\partial x}}}
 
,
among other possibilities.<a class="footnote-ref" id="fnref:41" href="#fn:41">41</a> It can be thought of as the rate of change of the function in the 
 
 
 
 x
 
 
 {\displaystyle x}
 
-direction.<a class="footnote-ref" id="fnref:42" href="#fn:42">42</a> Here <a href="/facts/%25E2%2588%2582/1SECcBj2">∂</a> is a rounded d called the partial derivative symbol. To distinguish it from the letter d, ∂ is sometimes pronounced "der", "del", or "partial" instead of "dee".<a class="footnote-ref" id="fnref:43" href="#fn:43">43</a> For example, let ⁠
 
 
 
 f
 (
 x
 ,
 y
 )
 =
 
 x
 
 2
 
 
 +
 x
 y
 +
 
 y
 
 2
 
 
 
 
 {\displaystyle f(x,y)=x^{2}+xy+y^{2}}
 
⁠, then the partial derivative of function 
 
 
 
 f
 
 
 {\displaystyle f}
 
 with respect to both variables 
 
 
 
 x
 
 
 {\displaystyle x}
 
 and 
 
 
 
 y
 
 
 {\displaystyle y}
 
 are, respectively:

∂
              f
            
            
              ∂
              x
            
          
        
        =
        2
        x
        +
        y
        ,
        
        
          
            
              ∂
              f
            
            
              ∂
              y
            
          
        
        =
        x
        +
        2
        y
        .
      
    
    {\displaystyle {\frac {\partial f}{\partial x}}=2x+y,\qquad {\frac {\partial f}{\partial y}}=x+2y.}

In general, the partial derivative of a function 
 
 
 
 f
 (
 
 x
 
 1
 
 
 ,
 …
 ,
 
 x
 
 n
 
 
 )
 
 
 {\displaystyle f(x_{1},\dots ,x_{n})}
 
 in the direction 
 
 
 
 
 x
 
 i
 
 
 
 
 {\displaystyle x_{i}}
 
 at the point 
 
 
 
 (
 
 a
 
 1
 
 
 ,
 …
 ,
 
 a
 
 n
 
 
 )
 
 
 {\displaystyle (a_{1},\dots ,a_{n})}
 
 is defined to be:<a class="footnote-ref" id="fnref:44" href="#fn:44">44</a>

∂
              f
            
            
              ∂
              
                x
                
                  i
                
              
            
          
        
        (
        
          a
          
            1
          
        
        ,
        …
        ,
        
          a
          
            n
          
        
        )
        =
        
          lim
          
            h
            →
            0
          
        
        
          
            
              f
              (
              
                a
                
                  1
                
              
              ,
              …
              ,
              
                a
                
                  i
                
              
              +
              h
              ,
              …
              ,
              
                a
                
                  n
                
              
              )
              −
              f
              (
              
                a
                
                  1
                
              
              ,
              …
              ,
              
                a
                
                  i
                
              
              ,
              …
              ,
              
                a
                
                  n
                
              
              )
            
            h
          
        
        .
      
    
    {\displaystyle {\frac {\partial f}{\partial x_{i}}}(a_{1},\ldots ,a_{n})=\lim _{h\to 0}{\frac {f(a_{1},\ldots ,a_{i}+h,\ldots ,a_{n})-f(a_{1},\ldots ,a_{i},\ldots ,a_{n})}{h}}.}

This is fundamental for the study of the <a href="/facts/Functions_of_several_real_variables/cyujR2q2">functions of several real variables</a>. Let 
 
 
 
 f
 (
 
 x
 
 1
 
 
 ,
 …
 ,
 
 x
 
 n
 
 
 )
 
 
 {\displaystyle f(x_{1},\dots ,x_{n})}
 
 be such a <a href="/facts/Real-valued_function/jetqlJpK">real-valued function</a>. If all partial derivatives 
 
 
 
 f
 
 
 {\displaystyle f}
 
 with respect to 
 
 
 
 
 x
 
 j
 
 
 
 
 {\displaystyle x_{j}}
 
 are defined at the point ⁠
 
 
 
 (
 
 a
 
 1
 
 
 ,
 …
 ,
 
 a
 
 n
 
 
 )
 
 
 {\displaystyle (a_{1},\dots ,a_{n})}
 
⁠, these partial derivatives define the vector

∇
        f
        (
        
          a
          
            1
          
        
        ,
        …
        ,
        
          a
          
            n
          
        
        )
        =
        
          (
          
            
              
                
                  ∂
                  f
                
                
                  ∂
                  
                    x
                    
                      1
                    
                  
                
              
            
            (
            
              a
              
                1
              
            
            ,
            …
            ,
            
              a
              
                n
              
            
            )
            ,
            …
            ,
            
              
                
                  ∂
                  f
                
                
                  ∂
                  
                    x
                    
                      n
                    
                  
                
              
            
            (
            
              a
              
                1
              
            
            ,
            …
            ,
            
              a
              
                n
              
            
            )
          
          )
        
        ,
      
    
    {\displaystyle \nabla f(a_{1},\ldots ,a_{n})=\left({\frac {\partial f}{\partial x_{1}}}(a_{1},\ldots ,a_{n}),\ldots ,{\frac {\partial f}{\partial x_{n}}}(a_{1},\ldots ,a_{n})\right),}

which is called the <a href="/facts/Gradient/TsR7uukj">gradient</a> of 
 
 
 
 f
 
 
 {\displaystyle f}
 
 at 
 
 
 
 a
 
 
 {\displaystyle a}
 
. If 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is differentiable at every point in some domain, then the gradient is a <a href="/facts/Vector-valued_function/qxu8aapt">vector-valued function</a> 
 
 
 
 ∇
 f
 
 
 {\displaystyle \nabla f}
 
 that maps the point 
 
 
 
 (
 
 a
 
 1
 
 
 ,
 …
 ,
 
 a
 
 n
 
 
 )
 
 
 {\displaystyle (a_{1},\dots ,a_{n})}
 
 to the vector 
 
 
 
 ∇
 f
 (
 
 a
 
 1
 
 
 ,
 …
 ,
 
 a
 
 n
 
 
 )
 
 
 {\displaystyle \nabla f(a_{1},\dots ,a_{n})}
 
. Consequently, the gradient determines a <a href="/facts/Vector_field/ccFNuPPp">vector field</a>.<a class="footnote-ref" id="fnref:45" href="#fn:45">45</a>

<h3>Directional derivatives</h3>
Main article: <a href="/facts/Directional_derivative/Fjc3JSVn">Directional derivative</a>
If 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is a real-valued function on ⁠
 
 
 
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {R} ^{n}}
 
⁠, then the partial derivatives of 
 
 
 
 f
 
 
 {\displaystyle f}
 
 measure its variation in the direction of the coordinate axes. For example, if 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is a function of 
 
 
 
 x
 
 
 {\displaystyle x}
 
 and ⁠
 
 
 
 y
 
 
 {\displaystyle y}
 
⁠, then its partial derivatives measure the variation in 
 
 
 
 f
 
 
 {\displaystyle f}
 
 in the 
 
 
 
 x
 
 
 {\displaystyle x}
 
 and 
 
 
 
 y
 
 
 {\displaystyle y}
 
 direction. However, they do not directly measure the variation of 
 
 
 
 f
 
 
 {\displaystyle f}
 
 in any other direction, such as along the diagonal line ⁠
 
 
 
 y
 =
 x
 
 
 {\displaystyle y=x}
 
⁠. These are measured using directional derivatives. Given a vector ⁠
 
 
 
 
 v
 
 =
 (
 
 v
 
 1
 
 
 ,
 …
 ,
 
 v
 
 n
 
 
 )
 
 
 {\displaystyle \mathbf {v} =(v_{1},\ldots ,v_{n})}
 
⁠, then the <a href="/facts/Directional_derivative/Fjc3JSVn">directional derivative</a> of 
 
 
 
 f
 
 
 {\displaystyle f}
 
 in the direction of 
 
 
 
 
 v
 
 
 
 {\displaystyle \mathbf {v} }
 
 at the point 
 
 
 
 
 x
 
 
 
 {\displaystyle \mathbf {x} }
 
 is:<a class="footnote-ref" id="fnref:46" href="#fn:46">46</a>

D
          
            
              v
            
          
        
        
          f
        
        (
        
          x
        
        )
        =
        
          lim
          
            h
            →
            0
          
        
        
          
            
              f
              (
              
                x
              
              +
              h
              
                v
              
              )
              −
              f
              (
              
                x
              
              )
            
            h
          
        
        .
      
    
    {\displaystyle D_{\mathbf {v} }{f}(\mathbf {x} )=\lim _{h\rightarrow 0}{\frac {f(\mathbf {x} +h\mathbf {v} )-f(\mathbf {x} )}{h}}.}

If all the partial derivatives of 
 
 
 
 f
 
 
 {\displaystyle f}
 
 exist and are continuous at ⁠
 
 
 
 
 x
 
 
 
 {\displaystyle \mathbf {x} }
 
⁠, then they determine the directional derivative of 
 
 
 
 f
 
 
 {\displaystyle f}
 
 in the direction 
 
 
 
 
 v
 
 
 
 {\displaystyle \mathbf {v} }
 
 by the formula:<a class="footnote-ref" id="fnref:47" href="#fn:47">47</a>

D
          
            
              v
            
          
        
        
          f
        
        (
        
          x
        
        )
        =
        
          ∑
          
            j
            =
            1
          
          
            n
          
        
        
          v
          
            j
          
        
        
          
            
              ∂
              f
            
            
              ∂
              
                x
                
                  j
                
              
            
          
        
        .
      
    
    {\displaystyle D_{\mathbf {v} }{f}(\mathbf {x} )=\sum _{j=1}^{n}v_{j}{\frac {\partial f}{\partial x_{j}}}.}

<h3>Total derivative and Jacobian matrix</h3>
Main article: <a href="/facts/Total_derivative/cPwsm9Wf">Total derivative</a>
When 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is a function from an open subset of 
 
 
 
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {R} ^{n}}
 
 to ⁠
 
 
 
 
 
 R
 
 
 m
 
 
 
 
 {\displaystyle \mathbb {R} ^{m}}
 
⁠, then the directional derivative of 
 
 
 
 f
 
 
 {\displaystyle f}
 
 in a chosen direction is the best linear approximation to 
 
 
 
 f
 
 
 {\displaystyle f}
 
 at that point and in that direction. However, when ⁠
 
 
 
 n
 >
 1
 
 
 {\displaystyle n>1}
 
⁠, no single directional derivative can give a complete picture of the behavior of 
 
 
 
 f
 
 
 {\displaystyle f}
 
. The total derivative gives a complete picture by considering all directions at once. That is, for any vector 
 
 
 
 
 v
 
 
 
 {\displaystyle \mathbf {v} }
 
 starting at ⁠
 
 
 
 
 a
 
 
 
 {\displaystyle \mathbf {a} }
 
⁠, the linear approximation formula holds:<a class="footnote-ref" id="fnref:48" href="#fn:48">48</a>

f
        (
        
          a
        
        +
        
          v
        
        )
        ≈
        f
        (
        
          a
        
        )
        +
        
          f
          ′
        
        (
        
          a
        
        )
        
          v
        
        .
      
    
    {\displaystyle f(\mathbf {a} +\mathbf {v} )\approx f(\mathbf {a} )+f'(\mathbf {a} )\mathbf {v} .}

Similarly with the single-variable derivative, 
 
 
 
 
 f
 ′
 
 (
 
 a
 
 )
 
 
 {\displaystyle f'(\mathbf {a} )}
 
 is chosen so that the error in this approximation is as small as possible. The total derivative of 
 
 
 
 f
 
 
 {\displaystyle f}
 
 at 
 
 
 
 
 a
 
 
 
 {\displaystyle \mathbf {a} }
 
 is the unique linear transformation 
 
 
 
 
 f
 ′
 
 (
 
 a
 
 )
 :
 
 
 R
 
 
 n
 
 
 →
 
 
 R
 
 
 m
 
 
 
 
 {\displaystyle f'(\mathbf {a} )\colon \mathbb {R} ^{n}\to \mathbb {R} ^{m}}
 
 such that<a class="footnote-ref" id="fnref:49" href="#fn:49">49</a>

lim
          
            
              h
            
            →
            0
          
        
        
          
            
              ‖
              f
              (
              
                a
              
              +
              
                h
              
              )
              −
              (
              f
              (
              
                a
              
              )
              +
              
                f
                ′
              
              (
              
                a
              
              )
              
                h
              
              )
              ‖
            
            
              ‖
              
                h
              
              ‖
            
          
        
        =
        0.
      
    
    {\displaystyle \lim _{\mathbf {h} \to 0}{\frac {\lVert f(\mathbf {a} +\mathbf {h} )-(f(\mathbf {a} )+f'(\mathbf {a} )\mathbf {h} )\rVert }{\lVert \mathbf {h} \rVert }}=0.}

Here 
 
 
 
 
 h
 
 
 
 {\displaystyle \mathbf {h} }
 
 is a vector in ⁠
 
 
 
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {R} ^{n}}
 
⁠, so the norm in the denominator is the standard length on 
 
 
 
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {R} ^{n}}
 
. However, 
 
 
 
 
 f
 ′
 
 (
 
 a
 
 )
 
 h
 
 
 
 {\displaystyle f'(\mathbf {a} )\mathbf {h} }
 
 is a vector in ⁠
 
 
 
 
 
 R
 
 
 m
 
 
 
 
 {\displaystyle \mathbb {R} ^{m}}
 
⁠, and the norm in the numerator is the standard length on 
 
 
 
 
 
 R
 
 
 m
 
 
 
 
 {\displaystyle \mathbb {R} ^{m}}
 
.<a class="footnote-ref" id="fnref:50" href="#fn:50">50</a> If 
 
 
 
 v
 
 
 {\displaystyle v}
 
 is a vector starting at ⁠
 
 
 
 a
 
 
 {\displaystyle a}
 
⁠, then 
 
 
 
 
 f
 ′
 
 (
 
 a
 
 )
 
 v
 
 
 
 {\displaystyle f'(\mathbf {a} )\mathbf {v} }
 
 is called the <a href="/facts/Pushforward_(differential)/xLsoLwVH">pushforward</a> of 
 
 
 
 
 v
 
 
 
 {\displaystyle \mathbf {v} }
 
 by 
 
 
 
 f
 
 
 {\displaystyle f}
 
.<a class="footnote-ref" id="fnref:51" href="#fn:51">51</a>
If the total derivative exists at ⁠
 
 
 
 
 a
 
 
 
 {\displaystyle \mathbf {a} }
 
⁠, then all the partial derivatives and directional derivatives of 
 
 
 
 f
 
 
 {\displaystyle f}
 
 exist at ⁠
 
 
 
 
 a
 
 
 
 {\displaystyle \mathbf {a} }
 
⁠, and for all ⁠
 
 
 
 
 v
 
 
 
 {\displaystyle \mathbf {v} }
 
⁠, 
 
 
 
 
 f
 ′
 
 (
 
 a
 
 )
 
 v
 
 
 
 {\displaystyle f'(\mathbf {a} )\mathbf {v} }
 
 is the directional derivative of 
 
 
 
 f
 
 
 {\displaystyle f}
 
 in the direction ⁠
 
 
 
 
 v
 
 
 
 {\displaystyle \mathbf {v} }
 
⁠. If 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is written using coordinate functions, so that ⁠
 
 
 
 f
 =
 (
 
 f
 
 1
 
 
 ,
 
 f
 
 2
 
 
 ,
 …
 ,
 
 f
 
 m
 
 
 )
 
 
 {\displaystyle f=(f_{1},f_{2},\dots ,f_{m})}
 
⁠, then the total derivative can be expressed using the partial derivatives as a <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a>. This matrix is called the <a href="/facts/Jacobian_matrix/0tNv6o8j">Jacobian matrix</a> of 
 
 
 
 f
 
 
 {\displaystyle f}
 
 at 
 
 
 
 
 a
 
 
 
 {\displaystyle \mathbf {a} }
 
:<a class="footnote-ref" id="fnref:52" href="#fn:52">52</a>

f
          ′
        
        (
        
          a
        
        )
        =
        
          Jac
          
            
              a
            
          
        
        =
        
          
            (
            
              
                
                  ∂
                  
                    f
                    
                      i
                    
                  
                
                
                  ∂
                  
                    x
                    
                      j
                    
                  
                
              
            
            )
          
          
            i
            j
          
        
        .
      
    
    {\displaystyle f'(\mathbf {a} )=\operatorname {Jac} _{\mathbf {a} }=\left({\frac {\partial f_{i}}{\partial x_{j}}}\right)_{ij}.}

<h2 id="generalizations">Generalizations</h2>
Main article: <a href="/facts/Generalizations_of_the_derivative/mgWY23Sd">Generalizations of the derivative</a>
The concept of a derivative can be extended to many other settings. The common thread is that the derivative of a function at a point serves as a <a href="/facts/Linear_approximation/SDBAneFx">linear approximation</a> of the function at that point.

<ul><li>An important generalization of the derivative concerns <a href="/facts/Complex_function/hgCoQlH0">complex functions</a> of <a href="/facts/Complex_number/2w7mqI7e">complex variables</a>, such as functions from (a domain in) the complex numbers 
 
 
 
 
 C
 
 
 
 {\displaystyle \mathbb {C} }
 
 to ⁠
 
 
 
 
 C
 
 
 
 {\displaystyle \mathbb {C} }
 
⁠. The notion of the derivative of such a function is obtained by replacing real variables with complex variables in the definition.<a class="footnote-ref" id="fnref:53" href="#fn:53">53</a> If 
 
 
 
 
 C
 
 
 
 {\displaystyle \mathbb {C} }
 
 is identified with 
 
 
 
 
 
 R
 
 
 2
 
 
 
 
 {\displaystyle \mathbb {R} ^{2}}
 
 by writing a complex number 
 
 
 
 z
 
 
 {\displaystyle z}
 
 as ⁠
 
 
 
 x
 +
 i
 y
 
 
 {\displaystyle x+iy}
 
⁠ then a differentiable function from 
 
 
 
 
 C
 
 
 
 {\displaystyle \mathbb {C} }
 
 to 
 
 
 
 
 C
 
 
 
 {\displaystyle \mathbb {C} }
 
 is certainly differentiable as a function from 
 
 
 
 
 
 R
 
 
 2
 
 
 
 
 {\displaystyle \mathbb {R} ^{2}}
 
 to 
 
 
 
 
 
 R
 
 
 2
 
 
 
 
 {\displaystyle \mathbb {R} ^{2}}
 
 (in the sense that its partial derivatives all exist), but the converse is not true in general: the complex derivative only exists if the real derivative is complex linear and this imposes relations between the partial derivatives called the <a href="/facts/Cauchy%25E2%2580%2593Riemann_equations/SWkIHflU">Cauchy–Riemann equations</a> – see <a href="/facts/Holomorphic_function/kVzthtEf">holomorphic functions</a>.<a class="footnote-ref" id="fnref:54" href="#fn:54">54</a></li>
<li>Another generalization concerns functions between <a href="/facts/Smooth_manifold/aSnbXKcV">differentiable or smooth manifolds</a>. Intuitively speaking such a manifold 
 
 
 
 M
 
 
 {\displaystyle M}
 
 is a space that can be approximated near each point 
 
 
 
 x
 
 
 {\displaystyle x}
 
 by a vector space called its <a href="/facts/Tangent_space/crvpbSeG">tangent space</a>: the prototypical example is a <a href="/facts/Smooth_surface/cslVnmd6">smooth surface</a> in ⁠
 
 
 
 
 
 R
 
 
 3
 
 
 
 
 {\displaystyle \mathbb {R} ^{3}}
 
⁠. The derivative (or differential) of a (differentiable) map 
 
 
 
 f
 :
 M
 →
 N
 
 
 {\displaystyle f:M\to N}
 
 between manifolds, at a point 
 
 
 
 x
 
 
 {\displaystyle x}
 
 in ⁠
 
 
 
 M
 
 
 {\displaystyle M}
 
⁠, is then a <a href="/facts/Linear_map/Q1PBKNVV">linear map</a> from the tangent space of 
 
 
 
 M
 
 
 {\displaystyle M}
 
 at 
 
 
 
 x
 
 
 {\displaystyle x}
 
 to the tangent space of 
 
 
 
 N
 
 
 {\displaystyle N}
 
 at ⁠
 
 
 
 f
 (
 x
 )
 
 
 {\displaystyle f(x)}
 
⁠. The derivative function becomes a map between the <a href="/facts/Tangent_bundle/dq8zW9Kb">tangent bundles</a> of 
 
 
 
 M
 
 
 {\displaystyle M}
 
 and ⁠
 
 
 
 N
 
 
 {\displaystyle N}
 
⁠. This definition is used in <a href="/facts/Differential_geometry/c04XmyLu">differential geometry</a>.<a class="footnote-ref" id="fnref:55" href="#fn:55">55</a></li>
<li>Differentiation can also be defined for maps between <a href="/facts/Vector_space/M1pxMLx2">vector space</a>, such as <a href="/facts/Banach_space/sDEIk7EC">Banach space</a>, in which those generalizations are the <a href="/facts/Gateaux_derivative/dqdtTtpf">Gateaux derivative</a> and the <a href="/facts/Fr%25C3%25A9chet_derivative/kyAlz3up">Fréchet derivative</a>.<a class="footnote-ref" id="fnref:56" href="#fn:56">56</a></li>
<li>One deficiency of the classical derivative is that very many functions are not differentiable. Nevertheless, there is a way of extending the notion of the derivative so that all <a href="/facts/Continuous_function/sKbl02pB">continuous</a> functions and many other functions can be differentiated using a concept known as the <a href="/facts/Weak_derivative/oehSmSMg">weak derivative</a>. The idea is to embed the continuous functions in a larger space called the space of <a href="/facts/Distribution_(mathematics)/yfaJ4mUp">distributions</a> and only require that a function is differentiable "on average".<a class="footnote-ref" id="fnref:57" href="#fn:57">57</a></li>
<li>Properties of the derivative have inspired the introduction and study of many similar objects in algebra and topology; an example is <a href="/facts/Differential_algebra/CXbpIMgk">differential algebra</a>. Here, it consists of the derivation of some topics in abstract algebra, such as <a href="/facts/Ring_(mathematics)/JnCUXz63">rings</a>, <a href="/facts/Ideal_(ring_theory)/vCvytduX">ideals</a>, <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a>, and so on.<a class="footnote-ref" id="fnref:58" href="#fn:58">58</a></li>
<li>The discrete equivalent of differentiation is <a href="/facts/Finite_difference/aY1txFhj">finite differences</a>. The study of differential calculus is unified with the calculus of finite differences in <a href="/facts/Time_scale_calculus/9T4xVS0t">time scale calculus</a>.<a class="footnote-ref" id="fnref:59" href="#fn:59">59</a></li>
<li>The <a href="/facts/Arithmetic_derivative/wfpUJko3">arithmetic derivative</a> involves the function that is defined for the <a href="/facts/Integer/PUwwrawx">integers</a> by the <a href="/facts/Prime_factorization/EjMCTz2t">prime factorization</a>. This is an analogy with the product rule.<a class="footnote-ref" id="fnref:60" href="#fn:60">60</a></li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Covariant_derivative/sRGN0Xh6">Covariant derivative</a></li>
<li><a href="/facts/Derivation_(differential_algebra)/vcSqpxzu">Derivation</a></li>
<li><a href="/facts/Exterior_derivative/p7HA0wze">Exterior derivative</a></li>
<li><a href="/facts/Functional_derivative/urNSIWYc">Functional derivative</a></li>
<li><a href="/facts/Integral/V7lyV997">Integral</a></li>
<li><a href="/facts/Lie_derivative/1KSJvrzk">Lie derivative </a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li><a href="/facts/Tom_M._Apostol/S6vGtJWN">Apostol, Tom M.</a> (June 1967), <a href="https://archive.org/details/calculus01apos">Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra</a>, vol. 1 (2nd ed.), Wiley, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-471-00005-1</li>
<li>Azegami, Hideyuki (2020), <a href="https://books.google.com/books?id=e08AEAAAQBAJ">Shape Optimization Problems</a>, Springer Optimization and Its Applications, vol. 164, Springer, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-981-15-7618-8">10.1007/978-981-15-7618-8</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-981-15-7618-8, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:226442409">226442409</a></li>
<li><a href="/facts/Stefan_Banach/b1qHSNg2">Banach, Stefan</a> (1931), <a href="https://scholar.google.com/scholar?output=instlink&q=info:SkKdCEmUd6QJ:scholar.google.com/&hl=en&as_sdt=0,50&scillfp=3432975470163241186&oi=lle">"Uber die Baire'sche Kategorie gewisser Funktionenmengen"</a>, Studia Math., 3 (3): 174–179, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.4064%2Fsm-3-1-174-179">10.4064/sm-3-1-174-179</a>.</li>
<li>Barbeau, E. J. (1961). <a href="https://doi.org/10.4153%2FCMB-1961-013-0">"Remarks on an arithmetic derivative"</a>. <a href="/facts/Canadian_Mathematical_Bulletin/BQ9Y16hL">Canadian Mathematical Bulletin</a>. 4 (2): 117–122. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.4153%2FCMB-1961-013-0">10.4153/CMB-1961-013-0</a>. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0101.03702">0101.03702</a>.</li>
<li>Bhardwaj, R. S. (2005), Mathematics for Economics & Business (2nd ed.), Excel Books India, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9788174464507</li>
<li><a href="/facts/Florian_Cajori/Yw4XQN0T">Cajori, Florian</a> (1923), "The History of Notations of the Calculus", Annals of Mathematics, 25 (1): 1–46, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F1967725">10.2307/1967725</a>, <a href="/facts/Hdl_(identifier)/rdebSxmC">hdl</a>:<a href="https://hdl.handle.net/2027%2Fmdp.39015017345896">2027/mdp.39015017345896</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/1967725">1967725</a></li>
<li>Cajori, Florian (2007), <a href="https://books.google.com/books?id=RUz1Us2UDh4C&pg=PA204">A History of Mathematical Notations</a>, vol. 2, Cosimo Classics, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-60206-713-4</li>
<li>Carothers, N. L. (2000), Real Analysis, Cambridge University Press</li>
<li>Choudary, A. D. R.; Niculescu, Constantin P. (2014), Real Analysis on Intervals, Springer India, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-81-322-2148-7">10.1007/978-81-322-2148-7</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-81-322-2148-7</li>
<li>Christopher, Essex (2013), Calculus: A complete course, Pearson, p. 682, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9780321781079, <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/872345701">872345701</a></li>
<li><a href="/facts/Richard_Courant/QDToNfWl">Courant, Richard</a>; <a href="/facts/Fritz_John/45S8VX2o">John, Fritz</a> (December 22, 1998), Introduction to Calculus and Analysis, Vol. 1, <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-1-4613-8955-2">10.1007/978-1-4613-8955-2</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-65058-4</li>
<li>David, Claire (2018), "Bypassing dynamical systems: A simple way to get the box-counting dimension of the graph of the Weierstrass function", Proceedings of the International Geometry Center, 11 (2), Academy of Sciences of Ukraine: 53–68, <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/1711.10349">1711.10349</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.15673%2Ftmgc.v11i2.1028">10.15673/tmgc.v11i2.1028</a></li>
<li>Davvaz, Bijan (2023), Vectors and Functions of Several Variables, Springer, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-981-99-2935-1">10.1007/978-981-99-2935-1</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-981-99-2935-1, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:259885793">259885793</a></li>
<li>Debnath, Lokenath; Shah, Firdous Ahmad (2015), <a href="https://books.google.com/books?id=qPuWBQAAQBAJ">Wavelet Transforms and Their Applications</a> (2nd ed.), Birkhäuser, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-0-8176-8418-1">10.1007/978-0-8176-8418-1</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8176-8418-1</li>
<li><a href="/facts/Lawrence_Evans/0XluMrne">Evans, Lawrence</a> (1999), Partial Differential Equations, American Mathematical Society, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8218-0772-2</li>
<li>Eves, Howard (January 2, 1990), An Introduction to the History of Mathematics (6th ed.), Brooks Cole, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-03-029558-4</li>
<li>Funaro, Daniele (1992), <a href="https://books.google.com/books?id=CX4SXf3mdeUC">Polynomial Approximation of Differential Equations</a>, Lecture Notes in Physics Monographs, vol. 8, Springer, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-3-540-46783-0">10.1007/978-3-540-46783-0</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-46783-0</li>
<li><a href="/facts/Greg_Gbur/KdKW5v3F">Gbur, Greg</a> (2011), Mathematical Methods for Optical Physics and Engineering, Cambridge University Press, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2011mmop.book.....G">2011mmop.book.....G</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-139-49269-0</li>
<li>Georgiev, Svetlin G. (2018), <a href="https://books.google.com/books?id=OJJVDwAAQBAJ">Fractional Dynamic Calculus and Fractional Dynamic Equations on Time Scales</a>, Springer, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-3-319-73954-0">10.1007/978-3-319-73954-0</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-319-73954-0</li>
<li>Goodman, A. W. (1963), <a href="https://books.google.com/books?id=v7i2YJJg_gcC">Analytic Geometry and the Calculus</a>, The MacMillan Company</li>
<li><a href="/facts/Larry_Gonick/mdlvsRko">Gonick, Larry</a> (2012), <a href="https://www.larrygonick.com/titles/science/cartoon-guide-to-calculus-2/">The Cartoon Guide to Calculus</a>, William Morrow, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-06-168909-3</li>
<li>Gray, Alfred; Abbena, Elsa; Salamon, Simon (2006), <a href="https://books.google.com/books?id=owEj9TMYo7IC">Modern Differential Geometry of Curves and Surfaces with Mathematica</a>, CRC Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-58488-448-4</li>
<li>Guzman, Alberto (2003), <a href="https://books.google.com/books?id=aI_qBwAAQBAJ">Derivatives and Integrals of Multivariable Functions</a>, Springer, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-1-4612-0035-2">10.1007/978-1-4612-0035-2</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4612-0035-2</li>
<li>Henle, James M.; Kleinberg, Eugene M. (2003), Infinitesimal Calculus, Dover Publications, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-42886-4</li>
<li><a href="/facts/Edwin_Hewitt/vY1ff529">Hewitt, Edwin</a>; Stromberg, Karl R. (1965), Real and abstract analysis, Springer-Verlag, Theorem 17.8, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-3-662-29794-0">10.1007/978-3-662-29794-0</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-662-28275-5</li>
<li>Jašek, Martin (1922), <a href="http://dml.cz/bitstream/handle/10338.dmlcz/121916/CasPestMatFys_051-1922-2_2.pdf">"Funkce Bolzanova"</a> (PDF), Časopis pro Pěstování Matematiky a Fyziky (in Czech), 51 (2): 69–76, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.21136%2FCPMF.1922.121916">10.21136/CPMF.1922.121916</a></li>
<li><a href="/facts/Vojt%25C4%259Bch_Jarn%25C3%25ADk/6SXVWszA">Jarník, Vojtěch</a> (1922), <a href="http://dml.cz/bitstream/handle/10338.dmlcz/109021/CasPestMatFys_051-1922-4_5.pdf">"O funkci Bolzanově"</a> (PDF), Časopis pro Pěstování Matematiky a Fyziky (in Czech), 51 (4): 248–264, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.21136%2FCPMF.1922.109021">10.21136/CPMF.1922.109021</a>. See the English version <a href="http://dml.cz/bitstream/handle/10338.dmlcz/400073/Bolzano_15-1981-1_6.pdf">here</a>.</li>
<li>Keisler, H. Jerome (2012) [1986], <a href="http://www.math.wisc.edu/~keisler/calc.html">Elementary Calculus: An Approach Using Infinitesimals</a> (2nd ed.), Prindle, Weber & Schmidt, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-871-50911-6</li>
<li>Kolchin, Ellis (1973), <a href="https://books.google.com/books?id=yDCfhIjka-8C">Differential Algebra And Algebraic Groups</a>, Academic Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-08-087369-5</li>
<li>Kreyszig, Erwin (1991), Differential Geometry, New York: <a href="/facts/Dover_Publications/RwjvlSFv">Dover</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-486-66721-9</li>
<li>Larson, Ron; Hostetler, Robert P.; Edwards, Bruce H. (February 28, 2006), Calculus: Early Transcendental Functions (4th ed.), Houghton Mifflin Company, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-618-60624-5</li>
<li>Lee, John M. (2013), Introduction to Smooth Manifolds, Graduate Texts in Mathematics, vol. 218, Springer, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-0-387-21752-9">10.1007/978-0-387-21752-9</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-21752-9</li>
<li>Mathai, A. M.; Haubold, H. J. (2017), <a href="https://books.google.com/books?id=v20uDwAAQBAJ">Fractional and Multivariable Calculus: Model Building and Optimization Problems</a>, Springer, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-3-319-59993-9">10.1007/978-3-319-59993-9</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-319-59993-9</li>
<li>Moore, Will H.; Siegel, David A. (2013), <a href="https://books.google.com/books?id=emmYDwAAQBAJ">A Mathematical Course for Political and Social Research</a>, Princeton University Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-691-15995-9</li>
<li>Roussos, Ioannis M. (2014), <a href="https://books.google.com/books?id=nX1cAgAAQBAJ">Improper Riemann Integral</a>, <a href="/facts/CRC_Press/duTuH4hz">CRC Press</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4665-8807-3</li>
<li>Rychlík, Karel (1923), Über eine Funktion aus Bolzanos handschriftlichem Nachlasse</li>
<li>Schwartzman, Steven (1994), <a href="https://books.google.com/books?id=PsH2DwAAQBAJ">The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English</a>, Mathematical Association of American, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9781614445012</li>
<li>Silverman, Richard A. (1989), Essential Calculus: With Applications, Courier Corporation, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9780486660974</li>
<li><a href="/facts/James_Stewart_(mathematician)/nPPwlFxt">Stewart, James</a> (December 24, 2002), <a href="https://archive.org/details/calculus0000stew">Calculus</a> (5th ed.), Brooks Cole, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-534-39339-7</li>
<li><a href="/facts/Gilbert_Strang/bbzaoPsm">Strang, Gilbert</a>; et al. (2023), <a href="https://openstax.org/details/books/calculus-volume-1">Calculus, volume 1</a>, OpenStax, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-947172-13-5</li>
<li><a href="/facts/George_B._Thomas/6HskCYmv">Thomas, George B. Jr.</a>; Weir, Maurice D.; <a href="/facts/Joel_Hass/9bxWBFNx">Hass, Joel</a> (2014). <a href="https://ia804708.us.archive.org/28/items/calculus.13th.edition-george.b.thomas/calculus.13th.edition-george.b.thomas.pdf">Thomas's Calculus</a> (PDF) (Thirteenth ed.). <a href="/facts/Pearson_PLC/Bo4IQzMo">Pearson PLC</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-321-87896-0. Retrieved September 9, 2024.</li>
<li><a href="/facts/Silvanus_P._Thompson/vPe7Daeu">Thompson, Silvanus P.</a> (September 8, 1998), <a href="/facts/Calculus_Made_Easy/RTaQvlV3">Calculus Made Easy</a> (Revised, Updated, Expanded ed.), New York: St. Martin's Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-312-18548-0</li>
<li>Varberg, Dale E.; Purcell, Edwin J.; Rigdon, Steven E. (2007), Calculus (9th ed.), <a href="/facts/Pearson_Prentice_Hall/bynWAuat">Pearson Prentice Hall</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0131469686</li>
<li><a href="/facts/Frank_Wilson_Warner/dfnYojtB">Warner, Frank W.</a> (1983), <a href="https://books.google.com/books?id=t6PNrjnfhuIC">Foundations of Differentiable Manifolds and Lie Groups</a>, Springer, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-90894-6</li></ul>

<h2 id="external-links">External links</h2>

<ul><li><a href="https://www.encyclopediaofmath.org/index.php?title=Derivative">"Derivative"</a>, <a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a>, 2001 [1994]</li>
<li><a href="/facts/Khan_Academy/4G99cy12">Khan Academy</a>: <a href="https://www.khanacademy.org/math/differential-calculus/taking-derivatives/intro_differential_calc/v/newton-leibniz-and-usain-bolt">"Newton, Leibniz, and Usain Bolt"</a></li>
<li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/Derivative.html">"Derivative"</a>. <a href="/facts/MathWorld/yc1jodbq">MathWorld</a>.</li>
<li><a href="http://www.wolframalpha.com/calculators/derivative-calculator/">Online Derivative Calculator</a> from <a href="/facts/Wolfram_Alpha/CIBhWgXA">Wolfram Alpha</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Apostol 1967, p. 160; Stewart 2002, pp. 129–130; Strang et al. 2023, p. 224. - Apostol, Tom M. (June 1967), Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra, vol. 1 (2nd ed.), Wiley, ISBN 978-0-471-00005-1 <a href="https://archive.org/details/calculus01apos" target="_blank">https://archive.org/details/calculus01apos</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Apostol 1967, p. 160; Stewart 2002, p. 127; Strang et al. 2023, p. 220. - Apostol, Tom M. (June 1967), Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra, vol. 1 (2nd ed.), Wiley, ISBN 978-0-471-00005-1 <a href="https://archive.org/details/calculus01apos" target="_blank">https://archive.org/details/calculus01apos</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Gonick 2012, p. 83; Thomas et al. 2014, p. 60. - Gonick, Larry (2012), The Cartoon Guide to Calculus, William Morrow, ISBN 978-0-06-168909-3 <a href="https://www.larrygonick.com/titles/science/cartoon-guide-to-calculus-2/" target="_blank">https://www.larrygonick.com/titles/science/cartoon-guide-to-calculus-2/</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Gonick 2012, p. 88; Strang et al. 2023, p. 234. - Gonick, Larry (2012), The Cartoon Guide to Calculus, William Morrow, ISBN 978-0-06-168909-3 <a href="https://www.larrygonick.com/titles/science/cartoon-guide-to-calculus-2/" target="_blank">https://www.larrygonick.com/titles/science/cartoon-guide-to-calculus-2/</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Gonick 2012, p. 83; Strang et al. 2023, p. 232. - Gonick, Larry (2012), The Cartoon Guide to Calculus, William Morrow, ISBN 978-0-06-168909-3 <a href="https://www.larrygonick.com/titles/science/cartoon-guide-to-calculus-2/" target="_blank">https://www.larrygonick.com/titles/science/cartoon-guide-to-calculus-2/</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Gonick 2012, pp. 77–80. - Gonick, Larry (2012), The Cartoon Guide to Calculus, William Morrow, ISBN 978-0-06-168909-3 <a href="https://www.larrygonick.com/titles/science/cartoon-guide-to-calculus-2/" target="_blank">https://www.larrygonick.com/titles/science/cartoon-guide-to-calculus-2/</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Thompson 1998, pp. 34, 104; Stewart 2002, p. 128. - Thompson, Silvanus P. (September 8, 1998), Calculus Made Easy (Revised, Updated, Expanded ed.), New York: St. Martin's Press, ISBN 978-0-312-18548-0 <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Thompson 1998, pp. 84–85. - Thompson, Silvanus P. (September 8, 1998), Calculus Made Easy (Revised, Updated, Expanded ed.), New York: St. Martin's Press, ISBN 978-0-312-18548-0 <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Keisler 2012, pp. 902–904. - Keisler, H. Jerome (2012) [1986], Elementary Calculus: An Approach Using Infinitesimals (2nd ed.), Prindle, Weber & Schmidt, ISBN 978-0-871-50911-6 <a href="http://www.math.wisc.edu/~keisler/calc.html" target="_blank">http://www.math.wisc.edu/~keisler/calc.html</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Keisler 2012, p. 45; Henle & Kleinberg 2003, p. 66. - Keisler, H. Jerome (2012) [1986], Elementary Calculus: An Approach Using Infinitesimals (2nd ed.), Prindle, Weber & Schmidt, ISBN 978-0-871-50911-6 <a href="http://www.math.wisc.edu/~keisler/calc.html" target="_blank">http://www.math.wisc.edu/~keisler/calc.html</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Gonick 2012, p. 156; Thomas et al. 2014, p. 114; Strang et al. 2023, p. 237. - Gonick, Larry (2012), The Cartoon Guide to Calculus, William Morrow, ISBN 978-0-06-168909-3 <a href="https://www.larrygonick.com/titles/science/cartoon-guide-to-calculus-2/" target="_blank">https://www.larrygonick.com/titles/science/cartoon-guide-to-calculus-2/</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Gonick 2012, p. 149; Thomas et al. 2014, p. 113; Strang et al. 2023, p. 237. - Gonick, Larry (2012), The Cartoon Guide to Calculus, William Morrow, ISBN 978-0-06-168909-3 <a href="https://www.larrygonick.com/titles/science/cartoon-guide-to-calculus-2/" target="_blank">https://www.larrygonick.com/titles/science/cartoon-guide-to-calculus-2/</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Gonick 2012, p. 156; Thomas et al. 2014, p. 114; Strang et al. 2023, pp. 237–238. - Gonick, Larry (2012), The Cartoon Guide to Calculus, William Morrow, ISBN 978-0-06-168909-3 <a href="https://www.larrygonick.com/titles/science/cartoon-guide-to-calculus-2/" target="_blank">https://www.larrygonick.com/titles/science/cartoon-guide-to-calculus-2/</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Jašek 1922; Jarník 1922; Rychlík 1923. - Jašek, Martin (1922), "Funkce Bolzanova" (PDF), Časopis pro Pěstování Matematiky a Fyziky (in Czech), 51 (2): 69–76, doi:10.21136/CPMF.1922.121916 <a href="http://dml.cz/bitstream/handle/10338.dmlcz/121916/CasPestMatFys_051-1922-2_2.pdf" target="_blank">http://dml.cz/bitstream/handle/10338.dmlcz/121916/CasPestMatFys_051-1922-2_2.pdf</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">David 2018. - David, Claire (2018), "Bypassing dynamical systems: A simple way to get the box-counting dimension of the graph of the Weierstrass function", Proceedings of the International Geometry Center, 11 (2), Academy of Sciences of Ukraine: 53–68, arXiv:1711.10349, doi:10.15673/tmgc.v11i2.1028 <a href="https://arxiv.org/abs/1711.10349" target="_blank">https://arxiv.org/abs/1711.10349</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Banach 1931, cited in Hewitt & Stromberg 1965. - Banach, Stefan (1931), "Uber die Baire'sche Kategorie gewisser Funktionenmengen", Studia Math., 3 (3): 174–179, doi:10.4064/sm-3-1-174-179. <a href="https://scholar.google.com/scholar?output=instlink&q=info:SkKdCEmUd6QJ:scholar.google.com/&hl=en&as_sdt=0,50&scillfp=3432975470163241186&oi=lle" target="_blank">https://scholar.google.com/scholar?output=instlink&q=info:SkKdCEmUd6QJ:scholar.google.com/&hl=en&as_sdt=0,50&scillfp=3432975470163241186&oi=lle</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">Apostol 1967, p. 172; Cajori 2007, p. 204. - Apostol, Tom M. (June 1967), Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra, vol. 1 (2nd ed.), Wiley, ISBN 978-0-471-00005-1 <a href="https://archive.org/details/calculus01apos" target="_blank">https://archive.org/details/calculus01apos</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
<li id="fn:18">Moore & Siegel 2013, p. 110. - Moore, Will H.; Siegel, David A. (2013), A Mathematical Course for Political and Social Research, Princeton University Press, ISBN 978-0-691-15995-9 <a href="https://books.google.com/books?id=emmYDwAAQBAJ" target="_blank">https://books.google.com/books?id=emmYDwAAQBAJ</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></li>
<li id="fn:19">Varberg, Purcell & Rigdon 2007, pp. 125–126. - Varberg, Dale E.; Purcell, Edwin J.; Rigdon, Steven E. (2007), Calculus (9th ed.), Pearson Prentice Hall, ISBN 978-0131469686 <a href="#fnref:19" class="footnote-back-ref">↩</a></li>
<li id="fn:20">In the formulation of calculus in terms of limits, various authors have assigned the 
 
 
 
 d
 u
 
 
 {\displaystyle du}
 
 symbol various meanings. Some authors such as Varberg, Purcell & Rigdon 2007, p. 119 and Stewart 2002, p. 177 do not assign a meaning to 
 
 
 
 d
 u
 
 
 {\displaystyle du}
 
 by itself, but only as part of the symbol 
 
 
 
 
 
 
 d
 u
 
 
 d
 x
 
 
 
 
 
 {\textstyle {\frac {du}{dx}}}
 
. Others define 
 
 
 
 d
 x
 
 
 {\displaystyle dx}
 
 as an independent variable, and define 
 
 
 
 d
 u
 
 
 {\displaystyle du}
 
 by ⁠
 
 
 
 
 d
 u
 =
 d
 x
 
 f
 ′
 
 (
 x
 )
 
 
 
 {\displaystyle \textstyle du=dxf'(x)}
 
⁠.
In non-standard analysis 
 
 
 
 d
 u
 
 
 {\displaystyle du}
 
 is defined as an infinitesimal. It is also interpreted as the exterior derivative of a function ⁠
 
 
 
 u
 
 
 {\displaystyle u}
 
⁠. See differential (infinitesimal) for further information. - Varberg, Dale E.; Purcell, Edwin J.; Rigdon, Steven E. (2007), Calculus (9th ed.), Pearson Prentice Hall, ISBN 978-0131469686 <a href="#fnref:20" class="footnote-back-ref">↩</a></li>
<li id="fn:21">Schwartzman 1994, p. 171; Cajori 1923, pp. 6–7, 10–12, 21–24. - Schwartzman, Steven (1994), The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English, Mathematical Association of American, ISBN 9781614445012 <a href="https://books.google.com/books?id=PsH2DwAAQBAJ" target="_blank">https://books.google.com/books?id=PsH2DwAAQBAJ</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></li>
<li id="fn:22">Moore & Siegel 2013, p. 110; Goodman 1963, pp. 78–79. - Moore, Will H.; Siegel, David A. (2013), A Mathematical Course for Political and Social Research, Princeton University Press, ISBN 978-0-691-15995-9 <a href="https://books.google.com/books?id=emmYDwAAQBAJ" target="_blank">https://books.google.com/books?id=emmYDwAAQBAJ</a> <a href="#fnref:22" class="footnote-back-ref">↩</a></li>
<li id="fn:23">Varberg, Purcell & Rigdon 2007, pp. 125–126; Cajori 2007, p. 228. - Varberg, Dale E.; Purcell, Edwin J.; Rigdon, Steven E. (2007), Calculus (9th ed.), Pearson Prentice Hall, ISBN 978-0131469686 <a href="#fnref:23" class="footnote-back-ref">↩</a></li>
<li id="fn:24">Choudary & Niculescu 2014, p. 222; Apostol 1967, p. 171. - Choudary, A. D. R.; Niculescu, Constantin P. (2014), Real Analysis on Intervals, Springer India, doi:10.1007/978-81-322-2148-7, ISBN 978-81-322-2148-7 <a href="https://doi.org/10.1007%2F978-81-322-2148-7" target="_blank">https://doi.org/10.1007%2F978-81-322-2148-7</a> <a href="#fnref:24" class="footnote-back-ref">↩</a></li>
<li id="fn:25">Varberg, Purcell & Rigdon 2007, pp. 125–126. - Varberg, Dale E.; Purcell, Edwin J.; Rigdon, Steven E. (2007), Calculus (9th ed.), Pearson Prentice Hall, ISBN 978-0131469686 <a href="#fnref:25" class="footnote-back-ref">↩</a></li>
<li id="fn:26">Evans 1999, p. 63; Kreyszig 1991, p. 1. - Evans, Lawrence (1999), Partial Differential Equations, American Mathematical Society, ISBN 0-8218-0772-2 <a href="#fnref:26" class="footnote-back-ref">↩</a></li>
<li id="fn:27">Varberg, Purcell & Rigdon 2007, pp. 125–126. - Varberg, Dale E.; Purcell, Edwin J.; Rigdon, Steven E. (2007), Calculus (9th ed.), Pearson Prentice Hall, ISBN 978-0131469686 <a href="#fnref:27" class="footnote-back-ref">↩</a></li>
<li id="fn:28">Cajori 1923. - Cajori, Florian (1923), "The History of Notations of the Calculus", Annals of Mathematics, 25 (1): 1–46, doi:10.2307/1967725, hdl:2027/mdp.39015017345896, JSTOR 1967725 <a href="https://doi.org/10.2307%2F1967725" target="_blank">https://doi.org/10.2307%2F1967725</a> <a href="#fnref:28" class="footnote-back-ref">↩</a></li>
<li id="fn:29">Apostol 1967, p. 172; Varberg, Purcell & Rigdon 2007, pp. 125–126. - Apostol, Tom M. (June 1967), Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra, vol. 1 (2nd ed.), Wiley, ISBN 978-0-471-00005-1 <a href="https://archive.org/details/calculus01apos" target="_blank">https://archive.org/details/calculus01apos</a> <a href="#fnref:29" class="footnote-back-ref">↩</a></li>
<li id="fn:30">Apostol 1967, p. 160. - Apostol, Tom M. (June 1967), Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra, vol. 1 (2nd ed.), Wiley, ISBN 978-0-471-00005-1 <a href="https://archive.org/details/calculus01apos" target="_blank">https://archive.org/details/calculus01apos</a> <a href="#fnref:30" class="footnote-back-ref">↩</a></li>
<li id="fn:31">Varberg, Purcell & Rigdon 2007. See p. 133 for the power rule, pp. 115–116 for the trigonometric functions, p. 326 for the natural logarithm, pp. 338–339 for exponential with base ⁠
 
 
 
 e
 
 
 {\displaystyle e}
 
⁠, p. 343 for the exponential with base ⁠
 
 
 
 a
 
 
 {\displaystyle a}
 
⁠, p. 344 for the logarithm with base ⁠
 
 
 
 a
 
 
 {\displaystyle a}
 
⁠, and p. 369 for the inverse of trigonometric functions. - Varberg, Dale E.; Purcell, Edwin J.; Rigdon, Steven E. (2007), Calculus (9th ed.), Pearson Prentice Hall, ISBN 978-0131469686 <a href="#fnref:31" class="footnote-back-ref">↩</a></li>
<li id="fn:32">For constant rule and sum rule, see Apostol 1967, pp. 161, 164, respectively. For the product rule, quotient rule, and chain rule, see Varberg, Purcell & Rigdon 2007, pp. 111–112, 119, respectively. For the special case of the product rule, that is, the product of a constant and a function, see Varberg, Purcell & Rigdon 2007, pp. 108–109. - Apostol, Tom M. (June 1967), Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra, vol. 1 (2nd ed.), Wiley, ISBN 978-0-471-00005-1 <a href="https://archive.org/details/calculus01apos" target="_blank">https://archive.org/details/calculus01apos</a> <a href="#fnref:32" class="footnote-back-ref">↩</a></li>
<li id="fn:33">Apostol 1967, p. 160; Varberg, Purcell & Rigdon 2007, pp. 125–126. - Apostol, Tom M. (June 1967), Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra, vol. 1 (2nd ed.), Wiley, ISBN 978-0-471-00005-1 <a href="https://archive.org/details/calculus01apos" target="_blank">https://archive.org/details/calculus01apos</a> <a href="#fnref:33" class="footnote-back-ref">↩</a></li>
<li id="fn:34">Warner 1983, p. 5. - Warner, Frank W. (1983), Foundations of Differentiable Manifolds and Lie Groups, Springer, ISBN 978-0-387-90894-6 <a href="https://books.google.com/books?id=t6PNrjnfhuIC" target="_blank">https://books.google.com/books?id=t6PNrjnfhuIC</a> <a href="#fnref:34" class="footnote-back-ref">↩</a></li>
<li id="fn:35">Debnath & Shah 2015, p. 40. - Debnath, Lokenath; Shah, Firdous Ahmad (2015), Wavelet Transforms and Their Applications (2nd ed.), Birkhäuser, doi:10.1007/978-0-8176-8418-1, ISBN 978-0-8176-8418-1 <a href="https://books.google.com/books?id=qPuWBQAAQBAJ" target="_blank">https://books.google.com/books?id=qPuWBQAAQBAJ</a> <a href="#fnref:35" class="footnote-back-ref">↩</a></li>
<li id="fn:36">Carothers 2000, p. 176. - Carothers, N. L. (2000), Real Analysis, Cambridge University Press <a href="#fnref:36" class="footnote-back-ref">↩</a></li>
<li id="fn:37">Apostol 1967, p. 160. - Apostol, Tom M. (June 1967), Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra, vol. 1 (2nd ed.), Wiley, ISBN 978-0-471-00005-1 <a href="https://archive.org/details/calculus01apos" target="_blank">https://archive.org/details/calculus01apos</a> <a href="#fnref:37" class="footnote-back-ref">↩</a></li>
<li id="fn:38">Stewart 2002, p. 193. - Stewart, James (December 24, 2002), Calculus (5th ed.), Brooks Cole, ISBN 978-0-534-39339-7 <a href="https://archive.org/details/calculus0000stew" target="_blank">https://archive.org/details/calculus0000stew</a> <a href="#fnref:38" class="footnote-back-ref">↩</a></li>
<li id="fn:39">Stewart 2002, p. 893. - Stewart, James (December 24, 2002), Calculus (5th ed.), Brooks Cole, ISBN 978-0-534-39339-7 <a href="https://archive.org/details/calculus0000stew" target="_blank">https://archive.org/details/calculus0000stew</a> <a href="#fnref:39" class="footnote-back-ref">↩</a></li>
<li id="fn:40">Stewart 2002, p. 893. - Stewart, James (December 24, 2002), Calculus (5th ed.), Brooks Cole, ISBN 978-0-534-39339-7 <a href="https://archive.org/details/calculus0000stew" target="_blank">https://archive.org/details/calculus0000stew</a> <a href="#fnref:40" class="footnote-back-ref">↩</a></li>
<li id="fn:41">Stewart 2002, p. 947; Christopher 2013, p. 682. - Stewart, James (December 24, 2002), Calculus (5th ed.), Brooks Cole, ISBN 978-0-534-39339-7 <a href="https://archive.org/details/calculus0000stew" target="_blank">https://archive.org/details/calculus0000stew</a> <a href="#fnref:41" class="footnote-back-ref">↩</a></li>
<li id="fn:42">Stewart 2002, p. 949. - Stewart, James (December 24, 2002), Calculus (5th ed.), Brooks Cole, ISBN 978-0-534-39339-7 <a href="https://archive.org/details/calculus0000stew" target="_blank">https://archive.org/details/calculus0000stew</a> <a href="#fnref:42" class="footnote-back-ref">↩</a></li>
<li id="fn:43">Silverman 1989, p. 216; Bhardwaj 2005, See p. 6.4. - Silverman, Richard A. (1989), Essential Calculus: With Applications, Courier Corporation, ISBN 9780486660974 <a href="#fnref:43" class="footnote-back-ref">↩</a></li>
<li id="fn:44">Mathai & Haubold 2017, p. 52. - Mathai, A. M.; Haubold, H. J. (2017), Fractional and Multivariable Calculus: Model Building and Optimization Problems, Springer, doi:10.1007/978-3-319-59993-9, ISBN 978-3-319-59993-9 <a href="https://books.google.com/books?id=v20uDwAAQBAJ" target="_blank">https://books.google.com/books?id=v20uDwAAQBAJ</a> <a href="#fnref:44" class="footnote-back-ref">↩</a></li>
<li id="fn:45">Gbur 2011, pp. 36–37. - Gbur, Greg (2011), Mathematical Methods for Optical Physics and Engineering, Cambridge University Press, Bibcode:2011mmop.book.....G, ISBN 978-1-139-49269-0 <a href="https://ui.adsabs.harvard.edu/abs/2011mmop.book.....G" target="_blank">https://ui.adsabs.harvard.edu/abs/2011mmop.book.....G</a> <a href="#fnref:45" class="footnote-back-ref">↩</a></li>
<li id="fn:46">Varberg, Purcell & Rigdon 2007, p. 642. - Varberg, Dale E.; Purcell, Edwin J.; Rigdon, Steven E. (2007), Calculus (9th ed.), Pearson Prentice Hall, ISBN 978-0131469686 <a href="#fnref:46" class="footnote-back-ref">↩</a></li>
<li id="fn:47">Guzman 2003, p. 35. - Guzman, Alberto (2003), Derivatives and Integrals of Multivariable Functions, Springer, doi:10.1007/978-1-4612-0035-2, ISBN 978-1-4612-0035-2 <a href="https://books.google.com/books?id=aI_qBwAAQBAJ" target="_blank">https://books.google.com/books?id=aI_qBwAAQBAJ</a> <a href="#fnref:47" class="footnote-back-ref">↩</a></li>
<li id="fn:48">Davvaz 2023, p. 266. - Davvaz, Bijan (2023), Vectors and Functions of Several Variables, Springer, doi:10.1007/978-981-99-2935-1, ISBN 978-981-99-2935-1, S2CID 259885793 <a href="https://doi.org/10.1007%2F978-981-99-2935-1" target="_blank">https://doi.org/10.1007%2F978-981-99-2935-1</a> <a href="#fnref:48" class="footnote-back-ref">↩</a></li>
<li id="fn:49">Davvaz 2023, p. 266. - Davvaz, Bijan (2023), Vectors and Functions of Several Variables, Springer, doi:10.1007/978-981-99-2935-1, ISBN 978-981-99-2935-1, S2CID 259885793 <a href="https://doi.org/10.1007%2F978-981-99-2935-1" target="_blank">https://doi.org/10.1007%2F978-981-99-2935-1</a> <a href="#fnref:49" class="footnote-back-ref">↩</a></li>
<li id="fn:50">Davvaz 2023, p. 266. - Davvaz, Bijan (2023), Vectors and Functions of Several Variables, Springer, doi:10.1007/978-981-99-2935-1, ISBN 978-981-99-2935-1, S2CID 259885793 <a href="https://doi.org/10.1007%2F978-981-99-2935-1" target="_blank">https://doi.org/10.1007%2F978-981-99-2935-1</a> <a href="#fnref:50" class="footnote-back-ref">↩</a></li>
<li id="fn:51">Lee 2013, p. 72. - Lee, John M. (2013), Introduction to Smooth Manifolds, Graduate Texts in Mathematics, vol. 218, Springer, doi:10.1007/978-0-387-21752-9, ISBN 978-0-387-21752-9 <a href="https://doi.org/10.1007%2F978-0-387-21752-9" target="_blank">https://doi.org/10.1007%2F978-0-387-21752-9</a> <a href="#fnref:51" class="footnote-back-ref">↩</a></li>
<li id="fn:52">Davvaz 2023, p. 267. - Davvaz, Bijan (2023), Vectors and Functions of Several Variables, Springer, doi:10.1007/978-981-99-2935-1, ISBN 978-981-99-2935-1, S2CID 259885793 <a href="https://doi.org/10.1007%2F978-981-99-2935-1" target="_blank">https://doi.org/10.1007%2F978-981-99-2935-1</a> <a href="#fnref:52" class="footnote-back-ref">↩</a></li>
<li id="fn:53">Roussos 2014, p. 303. - Roussos, Ioannis M. (2014), Improper Riemann Integral, CRC Press, ISBN 978-1-4665-8807-3 <a href="https://books.google.com/books?id=nX1cAgAAQBAJ" target="_blank">https://books.google.com/books?id=nX1cAgAAQBAJ</a> <a href="#fnref:53" class="footnote-back-ref">↩</a></li>
<li id="fn:54">Gbur 2011, pp. 261–264. - Gbur, Greg (2011), Mathematical Methods for Optical Physics and Engineering, Cambridge University Press, Bibcode:2011mmop.book.....G, ISBN 978-1-139-49269-0 <a href="https://ui.adsabs.harvard.edu/abs/2011mmop.book.....G" target="_blank">https://ui.adsabs.harvard.edu/abs/2011mmop.book.....G</a> <a href="#fnref:54" class="footnote-back-ref">↩</a></li>
<li id="fn:55">Gray, Abbena & Salamon 2006, p. 826. - Gray, Alfred; Abbena, Elsa; Salamon, Simon (2006), Modern Differential Geometry of Curves and Surfaces with Mathematica, CRC Press, ISBN 978-1-58488-448-4 <a href="https://books.google.com/books?id=owEj9TMYo7IC" target="_blank">https://books.google.com/books?id=owEj9TMYo7IC</a> <a href="#fnref:55" class="footnote-back-ref">↩</a></li>
<li id="fn:56">Azegami 2020. See p. 209 for the Gateaux derivative, and p. 211 for the Fréchet derivative. - Azegami, Hideyuki (2020), Shape Optimization Problems, Springer Optimization and Its Applications, vol. 164, Springer, doi:10.1007/978-981-15-7618-8, ISBN 978-981-15-7618-8, S2CID 226442409 <a href="https://books.google.com/books?id=e08AEAAAQBAJ" target="_blank">https://books.google.com/books?id=e08AEAAAQBAJ</a> <a href="#fnref:56" class="footnote-back-ref">↩</a></li>
<li id="fn:57">Funaro 1992, pp. 84–85. - Funaro, Daniele (1992), Polynomial Approximation of Differential Equations, Lecture Notes in Physics Monographs, vol. 8, Springer, doi:10.1007/978-3-540-46783-0, ISBN 978-3-540-46783-0 <a href="https://books.google.com/books?id=CX4SXf3mdeUC" target="_blank">https://books.google.com/books?id=CX4SXf3mdeUC</a> <a href="#fnref:57" class="footnote-back-ref">↩</a></li>
<li id="fn:58">Kolchin 1973, pp. 58, 126. - Kolchin, Ellis (1973), Differential Algebra And Algebraic Groups, Academic Press, ISBN 978-0-08-087369-5 <a href="https://books.google.com/books?id=yDCfhIjka-8C" target="_blank">https://books.google.com/books?id=yDCfhIjka-8C</a> <a href="#fnref:58" class="footnote-back-ref">↩</a></li>
<li id="fn:59">Georgiev 2018, p. 8. - Georgiev, Svetlin G. (2018), Fractional Dynamic Calculus and Fractional Dynamic Equations on Time Scales, Springer, doi:10.1007/978-3-319-73954-0, ISBN 978-3-319-73954-0 <a href="https://books.google.com/books?id=OJJVDwAAQBAJ" target="_blank">https://books.google.com/books?id=OJJVDwAAQBAJ</a> <a href="#fnref:59" class="footnote-back-ref">↩</a></li>
<li id="fn:60">Barbeau 1961. - Barbeau, E. J. (1961). "Remarks on an arithmetic derivative". Canadian Mathematical Bulletin. 4 (2): 117–122. doi:10.4153/CMB-1961-013-0. Zbl 0101.03702. <a href="https://doi.org/10.4153%2FCMB-1961-013-0" target="_blank">https://doi.org/10.4153%2FCMB-1961-013-0</a> <a href="#fnref:60" class="footnote-back-ref">↩</a></li>
</ol>

Derivative open-in-new

Derivative