High School
SAT
ACT
University
MCAT
LSAT
DAT
Wize AP Calculus (BC) Textbook > Differentiation 2: Composite, Implicit, & Inverse Functions

Derivatives of Inverse Trig Functions

Derivatives of Logarithmic FunctionsPrevious Section
Higher Order DerivativesNext Section
Popular Courses
Find My Course
0:00 / 0:00

Derivatives of Inverse Trig Functions

The derivatives of the six inverse trigonometric functions are:

(arcsin⁡(x))′=11−x2       (arccos⁡(x))′=−11−x2 (arctan⁡(x))′=11+x2  (arcsec(x))′=1∣x∣x2−1 (arcCot(x))′=−11+x2(arcCsc(x))′=−1∣x∣x2−1\boxed{\quad (\arcsin(x))' =\frac{1}{\sqrt{1-x^2}}\qquad \ \ \ \ \ \ \hspace{0.25in}\ (\arccos(x))'= \frac{-1}{\sqrt{1-x^2}}\quad }\\ \\\text{ }\\ \boxed{\quad (\arctan(x))' =\frac{1}{{1+x^2}} \ \ \qquad\hspace{0.16in}(\arcsec (x))'=\frac{1}{|x|\sqrt{x^2-1}}\quad }\\ \\\text{ }\\ \boxed{\quad (\arccot(x))' =\frac{-1}{{1+x^2}}\qquad\hspace{0.03in}(\arccsc (x))'=\frac{-1}{|x|\sqrt{x^2-1}}\quad }










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 Inverse Trig Derivatives!

0:00 / 0:00

Example: Inverse Trig Derivative

Find the derivative of f(x)=arcsin⁡(3x2)f(x)=\arcsin(3x^2).

f′(x)=[11−(3x2)2]×[3(2x)]\displaystyle f'\left(x\right)=\left[\frac{1}{\sqrt{1-\left(3x^2\right)^2}}\right]\times\left[3\left(2x\right)\right]

f′(x)=6x1−9x4f'(x)=\displaystyle \frac{6x}{\sqrt{1-9x^4}}
checklist
Mark Yourself Question
  1. Grab a piece of paper and try this problem yourself.
  2. When you're done, check the "I have answered this question" box below.
  3. View the solution and report whether you got it right or wrong.
Find the derivative of f(x)=(arccos⁡x)2sec⁡x\displaystyle f\left(x\right)=\frac{\left(\arccos x\right)^2}{\sec x}