# 3.6 | Derivatives of Inverse Functions

The Derivative of an Inverse Function: Let $$f$$ be a differentiable function on an interval $$I$$. If $$f$$ has an inverse function $$g$$, then $$g$$ is differentiable at any $$x$$ for which $$f'(g(x)) \not = 0$$ and
$$g'(x) = \frac{1}{f'(g(x))}.$$
More Basic Differentiation Rules:

• $$\displaystyle \frac{d}{dx}[\ln x] = \frac{1}{x}$$
• $$\displaystyle \frac{d}{dx}[\log_a x] = \frac{1}{x \ln a}$$
• $$\displaystyle \frac{d}{dx}[\sin^{-1} x] = \frac{1}{\sqrt{1-x^2}}$$
• $$\displaystyle \frac{d}{dx}[\sec^{-1} x] = \frac{1}{|x|\sqrt{x^2-1}}$$
• $$\displaystyle \frac{d}{dx}[\cos^{-1} x] = \frac{-1}{\sqrt{1-x^2}}$$
• $$\displaystyle \frac{d}{dx}[\csc^{-1} x] = \frac{-1}{|x|\sqrt{x^2-1}}$$
• $$\displaystyle \frac{d}{dx}[\tan^{-1} x] = \frac{1}{1+x^2}$$
• $$\displaystyle \frac{d}{dx}[\cot^{-1} x] = \frac{-1}{1+x^2}$$