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:

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

E 3.6 Exercises