Learning Outcomes
- Understand the derivative of an inverse function.
- Calculate derivatives of the standard inverse trigonometric functions.
Textbook
- Chapter 3.7 Derivatives of Inverse Functions
Textbook Assignment
- p. 306: 265, 267, 279, 280, 281, 283
WeBWork Assignment
- Derivatives-Inverses
Exit problems of the session
-
Find the derivative of the following functions:
(a). $y=\cos^{-1}(\sqrt{x})$ (b). $y=x\tan^{-1}(3x)$ (c). $y=(1+\sin^{-1}x)^3$
Key Concepts
- Differentiation formulas for the inverse trigonometric functions.
$\dfrac{d}{dx}(\sin^{-1}x)=\dfrac{1}{\sqrt{1-x^2}}$
$\dfrac{d}{dx}(\cos^{-1}x)=\dfrac{-1}{\sqrt{1-x^2}}$
$\dfrac{d}{dx}(\tan^{-1}x)=\dfrac{1}{1+x^2}$
$\dfrac{d}{dx}(\cot^{-1}x)=\dfrac{-1}{1+x^2}$
$\dfrac{d}{dx}(\sec^{-1}x)=\dfrac{1}{|x|\sqrt{x^2-1}}$
$\dfrac{d}{dx}(\csc^{-1}x)=\dfrac{-1}{|x|\sqrt{x^2-1}}$
Videos and Practice Problems of Selected Topics
- $\rhd$ Derivatives of inverse functions from equation (5:03) Given $f(x) =\dfrac{1}{2}x^3+3x-4$ and let $h$ be the inverse function of $f$. Notice that $f(-2)=-14$. Find $h'(-14)$.
- $\rhd$ Derivatives of inverse sine function (4:55) Deriving the formula of inverse sine function, $\sin^{-1}x$.
- $\rhd$ Derivatives of inverse cosine function (3:43) Deriving the formula of inverse cosine function, $\cos^{-1}x$.
- $\rhd$ Derivatives of inverse tangent function (6:01) Deriving the formula of inverse tangent function, $\tan^{-1}x$.
- * Practice: Derivatives of inverse trigonometric functions. (4 problems)
