High School
SAT
ACT
University
MCAT
LSAT
DAT
Wize AP Calculus (BC) Textbook > Differentiation 1: Basics & Fundamental Properties

Derivatives of Trig Functions

The Quotient RulePrevious Section
Derivatives of Exponential FunctionsNext Section
Popular Courses
Find My Course
0:00 / 0:00

Derivatives of Trig Functions

We may often want to take derivatives involving the trigonometric functions. Rather than deriving them each time, it's much easier to simply remember the following derivatives.

Trig Derivatives

The derivatives of the six trigonometric functions are:

(sin⁡x)′=cos⁡x          (cos⁡x)′=−sin⁡x(tan⁡x)′=sec⁡2x      (sec⁡x)′=sec⁡xtan⁡x(cot⁡x)′=−csc⁡2x(csc⁡x)′=−csc⁡xcot⁡x\begin{array}{c} \boxed{(\sin x)'=\cos x}\ \ \ \ \ \hspace{1 cm}\ \ \ \ \ \boxed{(\cos x)'=-\sin x } \\ \\ \boxed{(\tan x)'=\sec^{2} x} \ \ \ \ \ \ \qquad\boxed{(\sec x)'=\sec x\tan x} \\ \\ \boxed{ (\cot x)'=-\csc^{2}x}\qquad\boxed{(\csc x)'=-\csc x\cot x } \end{array}


Wize Tip
Notice how all the derivatives of "CO Functions" have a minus sign.

Wize Concept
The Product, Quotient, and Chain Rules still apply for Trig Derivatives!

0:00 / 0:00

Example: Trig Derivatives


Find ddx(x2sin⁡x)\displaystyle\frac{d}{dx}(x^2\sin{x}).

ddx(x2sin⁡x)\displaystyle \frac{d}{dx}(x^2\sin{x})
=(ddxx2)×sin⁡x+x2×(ddxsin⁡x)\displaystyle =(\frac{d}{dx}x^2)\times\sin{x}+x^2\times(\frac{d}{dx}\sin{x})
=2xsin⁡x+x2cos⁡x\displaystyle =2x\sin{x}+x^2\cos{x}
0:00 / 0:00

Example: Trig Derivatives

Find the derivative of f(x)=1+cos⁡xx−sin⁡x\displaystyle f(x)=\frac{1+\cos x}{x-\sin x}


f(x)=y=1+cos⁡xx−sin⁡x\displaystyle f(x)=y=\frac{1+\cos x}{x-\sin x}

Quotient rule:

y′=−sin⁡x(x−sin⁡x)−(1+cos⁡x)(1−cos⁡x)(x−sin⁡x)2y'=\dfrac{-\sin x(x-\sin x)-(1+\cos x)(1-\cos x)}{(x-\sin x)^2}

=−xsin⁡x+sin⁡2x−1+cos⁡2x(x−sin⁡x)2=\dfrac{-x\sin x+\sin ^2x-1+\cos ^2x}{(x-\sin x)^2}

=−xsin⁡x(x−sin⁡x)2=-\dfrac{x\sin x}{(x-\sin x)^2}
Find the derivative of

f(θ)=(sin⁡θ−cos⁡θ)sec⁡θ\displaystyle f(\theta)=(\sin\theta-\cos\theta)\sec\theta



Find y(23)y^{(23)} , the 23rd derivatives, given y=sin⁡x\displaystyle y=\sin x.