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Reference.org
Differentiation of trigonometric functions
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<table><tbody><tr><th>Function</th><th>Derivative</th></tr><tr><td> sin ( x ) {\displaystyle \sin(x)} </td><td> cos ( x ) {\displaystyle \cos(x)} </td></tr><tr><td> cos ( x ) {\displaystyle \cos(x)} </td><td> − sin ( x ) {\displaystyle -\sin(x)} </td></tr><tr><td> tan ( x ) {\displaystyle \tan(x)} </td><td> sec 2 ( x ) {\displaystyle \sec ^{2}(x)} </td></tr><tr><td> cot ( x ) {\displaystyle \cot(x)} </td><td> − csc 2 ( x ) {\displaystyle -\csc ^{2}(x)} </td></tr><tr><td> sec ( x ) {\displaystyle \sec(x)} </td><td> sec ( x ) tan ( x ) {\displaystyle \sec(x)\tan(x)} </td></tr><tr><td> csc ( x ) {\displaystyle \csc(x)} </td><td> − csc ( x ) cot ( x ) {\displaystyle -\csc(x)\cot(x)} </td></tr><tr><td> arcsin ( x ) {\displaystyle \arcsin(x)} </td><td> 1 1 − x 2 {\displaystyle {\frac {1}{\sqrt {1-x^{2}}}}} </td></tr><tr><td> arccos ( x ) {\displaystyle \arccos(x)} </td><td> − 1 1 − x 2 {\displaystyle -{\frac {1}{\sqrt {1-x^{2}}}}} </td></tr><tr><td> arctan ( x ) {\displaystyle \arctan(x)} </td><td> 1 x 2 + 1 {\displaystyle {\frac {1}{x^{2}+1}}} </td></tr><tr><td> arccot ( x ) {\displaystyle \operatorname {arccot}(x)} </td><td> − 1 x 2 + 1 {\displaystyle -{\frac {1}{x^{2}+1}}} </td></tr><tr><td> arcsec ( x ) {\displaystyle \operatorname {arcsec}(x)} </td><td> 1 | x | x 2 − 1 {\displaystyle {\frac {1}{|x|{\sqrt {x^{2}-1}}}}} </td></tr><tr><td> arccsc ( x ) {\displaystyle \operatorname {arccsc}(x)} </td><td> − 1 | x | x 2 − 1 {\displaystyle -{\frac {1}{|x|{\sqrt {x^{2}-1}}}}} </td></tr></tbody></table> <p>The differentiation of trigonometric functions is the mathematical process of finding the <a href="/facts/Derivative/7Ito70Dh">derivative</a> of a <a href="/facts/Trigonometric_function/czej7ur0">trigonometric function</a>, or its rate of change with respect to a variable. For example, the derivative of the sine function is written sin′(<i>a</i>) = cos(<i>a</i>), meaning that the rate of change of sin(<i>x</i>) at a particular angle <i>x = a</i> is given by the cosine of that angle. </p><p>All derivatives of circular trigonometric functions can be found from those of sin(<i>x</i>) and cos(<i>x</i>) by means of the <a href="/facts/Quotient_rule/GBvTloJ5">quotient rule</a> applied to functions such as tan(<i>x</i>) = sin(<i>x</i>)/cos(<i>x</i>). Knowing these derivatives, the derivatives of the <a href="/facts/Inverse_trigonometric_functions/bVwU9FPG">inverse trigonometric functions</a> are found using <a href="/facts/Implicit_differentiation/UJmQ2Cyk">implicit differentiation</a>. </p>