Appendix → Derivatives of Trigonometric Functions Derivatives of Trigonometric Functions
Quick reference for the derivatives of the standard trigonometric functions. These follow from the differentiation rules combined with the limit definitions of sin \sin sin cos \cos cos
Standard Derivatives
The derivatives of the trigonometric functions are:
d d x sin x = cos x d d x cos x = − sin x d d x tan x = 1 cos 2 x = sec 2 x \frac{d}{dx} \sin x = \cos x \qquad \frac{d}{dx} \cos x = -\sin x \qquad \frac{d}{dx} \tan x = \frac{1}{\cos^2 x} = \sec^2 x d x d sin x = cos x d x d cos x = − sin x d x d tan x = c o s 2 x 1 = sec 2 x
d d x cot x = − 1 sin 2 x = − csc 2 x d d x sec x = sec x tan x d d x csc x = − csc x cot x \frac{d}{dx} \cot x = -\frac{1}{\sin^2 x} = -\csc^2 x \qquad \frac{d}{dx} \sec x = \sec x \tan x \qquad \frac{d}{dx} \csc x = -\csc x \cot x d x d cot x = − s i n 2 x 1 = − csc 2 x d x d sec x = sec x tan x d x d csc x = − csc x cot x
Inverse Trigonometric Functions
d d x arcsin x = 1 1 − x 2 d d x arccos x = − 1 1 − x 2 d d x arctan x = 1 1 + x 2 \frac{d}{dx} \arcsin x = \frac{1}{\sqrt{1 - x^2}} \qquad \frac{d}{dx} \arccos x = -\frac{1}{\sqrt{1 - x^2}} \qquad \frac{d}{dx} \arctan x = \frac{1}{1 + x^2} d x d arcsin x = 1 − x 2 1 d x d arccos x = − 1 − x 2 1 d x d arctan x = 1 + x 2 1
With the Chain Rule
These become the building blocks for differentiating composite expressions. For a differentiable function g ( x ) g(x) g ( x )
d d x sin ( g ( x ) ) = cos ( g ( x ) ) ⋅ g ′ ( x ) \frac{d}{dx} \sin(g(x)) = \cos(g(x)) \cdot g'(x) d x d sin ( g ( x )) = cos ( g ( x )) ⋅ g ′ ( x )
d d x cos ( g ( x ) ) = − sin ( g ( x ) ) ⋅ g ′ ( x ) \frac{d}{dx} \cos(g(x)) = -\sin(g(x)) \cdot g'(x) d x d cos ( g ( x )) = − sin ( g ( x )) ⋅ g ′ ( x )
d d x sin ( 3 x 2 ) = cos ( 3 x 2 ) ⋅ 6 x = 6 x cos ( 3 x 2 ) \dfrac{d}{dx} \sin(3x^2) = \cos(3x^2) \cdot 6x = 6x \cos(3x^2) d x d sin ( 3 x 2 ) = cos ( 3 x 2 ) ⋅ 6 x = 6 x cos ( 3 x 2 )
d d z e − z sin x = − e − z sin x \dfrac{d}{dz} e^{-z} \sin x = -e^{-z} \sin x d z d e − z sin x = − e − z sin x