3.6 | The Chain Rule and Inverse Functions


Derivative of \(a^x\):
$$ \frac{d}{dx}[a^x] = a^x \ln a$$
Differentiation Rules of Some Inverse Functions:

  • \( \displaystyle \frac{d}{dx}[\ln x] = \frac{1}{x}\)
  • \( \displaystyle \frac{d}{dx}[\log_a x] = \frac{1}{x \ln a}\)
  • \( \displaystyle \frac{d}{dx}[\sin^{-1} x] = \frac{1}{\sqrt{1-x^2}}\)
  • \( \displaystyle \frac{d}{dx}[\sec^{-1} x] = \frac{1}{|x|\sqrt{x^2-1}}\)
  • \( \displaystyle \frac{d}{dx}[\cos^{-1} x] = \frac{-1}{\sqrt{1-x^2}}\)
  • \( \displaystyle \frac{d}{dx}[\csc^{-1} x] = \frac{-1}{|x|\sqrt{x^2-1}}\)
  • \( \displaystyle \frac{d}{dx}[\tan^{-1} x] = \frac{1}{1+x^2}\)
  • \( \displaystyle \frac{d}{dx}[\cot^{-1} x] = \frac{-1}{1+x^2}\)
Derivative of a General Inverse Function:
$$
\frac{d}{dx}[f^{-1}(x)] = \frac{1}{f'(f^{-1}(x))}
$$

E 3.6 Exercises