Chapter 4, Problem 17RE

To determine

To Find: The derivative of the given function.

Answer to Problem 17RE

The derivative is $\frac{-1}{{\cos }^{-1}x\sqrt{1-x^2}}$ .

Explanation of Solution

Given information: Given function $y=\ln ({\cos }^{-1}x)$

Formula used: $\frac{d(\ln x)}{dx}=\frac{1}{x}\text{and}\frac{d({\cos }^{-1}x)}{dx}=\frac{-1}{\sqrt{1-x^2}}$ with chain rule.

Calculation:

  $\begin{array}{l}y=\ln ({\cos }^{-1}x)\\ \text{Differentiate}\text{with}\text{respect}\text{to}x\\ \frac{dy}{dx}=\frac{d\ln ({\cos }^{-1}x)}{d({\cos }^{-1}x)}\times \frac{d({\cos }^{-1}x)}{dx}\\ \frac{dy}{dx}=\frac{1}{{\cos }^{-1}x}\times \frac{-1}{\sqrt{1-x^2}}\\ \frac{dy}{dx}=\frac{-1}{{\cos }^{-1}x\sqrt{1-x^2}}\end{array}$