Calculus Reference

Trigonometric Derivatives

Differentiate sine, cosine, tangent, and the reciprocal trigonometric functions. Then use the chain rule when the angle is an expression.

Solved Examples

Begin with the basic derivatives, then practice a composite angle and repeated differentiation.

Differentiate sine
Differentiate cosine
Differentiate tangent from sine and cosine
Apply the chain rule to a trig function
Find the first three derivatives of sine

Trigonometric Derivative Table

Function	Derivative	Where the function is defined
$\sin x$	$\cos x$	All real $x$.
$\cos x$	$-\sin x$	All real $x$.
$\tan x$	$\sec^2x$	$\cos x\ne0$.
$\csc x$	$-\csc x\cot x$	$\sin x\ne0$.
$\sec x$	$\sec x\tan x$	$\cos x\ne0$.
$\cot x$	$-\csc^2x$	$\sin x\ne0$.

These formulas assume angles are measured in radians. If the angle is in degrees, convert it to radians first or include the factor $\frac{\pi}{180}$ from the chain rule.

Chain Rule for Trigonometric Functions

For an inside function $u=u(x)$, keep $u$ in the trigonometric function and multiply by $u'(x)$.

Composite function	Derivative
$\sin(u)$	$\cos(u)u'$
$\cos(u)$	$-\sin(u)u'$
$\tan(u)$	$\sec^2(u)u'$
$\sec(u)$	$\sec(u)\tan(u)u'$

See the full chain rule lesson for the general pattern.

Example: Differentiate Sine

Find the Derivative of $f(x)=\sin x$

The basic sine rule changes sine to cosine.

Solution Steps

Apply $\frac{d}{dx}[\sin x]=\cos x$.

$$\boxed{f'(x)=\cos x}$$

Example: Differentiate Cosine

Find the Derivative of $g(x)=\cos x$

The cosine rule includes a negative sign.

Solution Steps

Apply $\frac{d}{dx}[\cos x]=-\sin x$.

$$\boxed{g'(x)=-\sin x}$$

Example: Derive the Tangent Rule

Because $\tan x=\frac{\sin x}{\cos x}$, the quotient rule gives the tangent derivative.

Differentiate $h(x)=\tan x$

Use the quotient rule and the identity $\sin^2x+\cos^2x=1$.

Solution Steps

Rewrite tangent as $\sin x/\cos x$ and apply the quotient rule.
Use the Pythagorean identity in the numerator.
Rewrite $1/\cos^2x$ as $\sec^2x$.

$$h'(x)=\frac{(\cos x)(\cos x)-(\sin x)(-\sin x)}{\cos^2x}$$$$h'(x)=\frac{\cos^2x+\sin^2x}{\cos^2x}$$$$\boxed{h'(x)=\sec^2x},\quad \cos x\ne0$$

Example: Trigonometric Chain Rule

Differentiate $p(x)=\sin(3x^2+1)$

The outside function is sine; the inside expression is $3x^2+1$.

Solution Steps

Differentiate the outside sine function and keep its inside.
Multiply by the inside derivative, $6x$.

$$p'(x)=\cos(3x^2+1)(6x)$$$$\boxed{p'(x)=6x\cos(3x^2+1)}$$

Example: First, Second, and Third Derivatives

Repeated derivatives of sine cycle through sine and cosine, with alternating signs.

Find Successive Derivatives of $q(x)=\sin x$

Solution Steps

Differentiate sine to find $q'(x)$.
Differentiate cosine to find $q''(x)$.
Differentiate negative sine to find $q'''(x)$.

$$q'(x)=\cos x$$$$q''(x)=-\sin x$$$$q'''(x)=-\cos x$$

For supported polynomial and rational forms, the derivative calculators can report successive derivatives through a selected order.

Continue Studying Derivatives

Review the special-function derivative guide, study exponential and logarithmic derivatives, or continue to inverse-trigonometric derivatives.