Chapter 3, Problem 119RE

Derivative of the inverse in two ways Let $f\left(x\right)=\sin x,f^{-1}\left(x\right)={\sin }^{-1}\left(x\right)$, and $\left(x_0,y_0\right)=\left(\pi /4,1/\sqrt{2}\right)$.

  a.    Evaluate ${\left(f^{-1}\right)}^{\prime }\left(1/\sqrt{2}\right)$ using Theorem 3.21.

  b.    Evaluate ${\left(f^{-1}\right)}^{\prime }\left(1/\sqrt{2}\right)$ directly by differentiating f^–1. Check for agreement with part (a).