Differentiable function

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a differentiable function of one <a href="/facts/Real_number/R02gw5Pb">real</a> variable is a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> whose <a href="/facts/Derivative/7Ito70Dh">derivative</a> exists at each point in its <a href="/facts/Domain_of_a_function/FQvoeUpX">domain</a>. In other words, the <a href="/facts/Graph_of_a_function/htYX0BGX">graph</a> of a differentiable function has a non-<a href="/facts/Vertical_tangent/GC5NTJEE">vertical</a> <a href="/facts/Tangent_line/sF0jH3EI">tangent line</a> at each interior point in its domain. A differentiable function is <a href="/facts/Smoothness/mffL4ch2">smooth</a> (the function is locally well approximated as a <a href="/facts/Linear_function/Ejvy8CKy">linear function</a> at each interior point) and does not contain any break, angle, or <a href="/facts/Cusp_(singularity)/QH3fEGry">cusp</a>.
If x0 is an interior point in the domain of a function f, then f is said to be differentiable at x0 if the derivative 
 
 
 
 
 f
 ′
 
 (
 
 x
 
 0
 
 
 )
 
 
 {\displaystyle f'(x_{0})}
 
 exists. In other words, the graph of f has a non-vertical tangent line at the point (x0, f(x0)). f is said to be differentiable on U if it is differentiable at every point of U. f is said to be continuously differentiable if its derivative is also a continuous function over the domain of the function 
 
 
 
 f
 
 
 {\textstyle f}
 
. Generally speaking, f is said to be of class 
 
 
 
 
 C
 
 k
 
 
 
 
 {\displaystyle C^{k}}
 
 if its first 
 
 
 
 k
 
 
 {\displaystyle k}
 
 derivatives 
 
 
 
 
 f
 
 ′
 
 
 (
 x
 )
 ,
 
 f
 
 ′
 ′
 
 
 (
 x
 )
 ,
 …
 ,
 
 f
 
 (
 k
 )
 
 
 (
 x
 )
 
 
 {\textstyle f^{\prime }(x),f^{\prime \prime }(x),\ldots ,f^{(k)}(x)}
 
 exist and are continuous over the domain of the function 
 
 
 
 f
 
 
 {\textstyle f}
 
.
For a multivariable function, as shown here, the differentiability of it is something more complex than the existence of the partial derivatives of it.

Differentiable function open-in-new

Differentiable function