Chapter 4.3, Problem 34E

To determine

To Find: The derivative of the given function.

Answer to Problem 34E

  $\frac{-1}{x\sqrt{x^2-1}}$

Explanation of Solution

Given information: Given function $y=\cos e{c}^{-1}x$

Formula used: $\frac{d(\cos ecx)}{dx}=-\cos ecx\cot x\text{,}\cos ec(\cos e{c}^{-1}x)=x\text{and}\cot x=\sqrt{\cos e{c}^{2}x-1}$ with chain rule.

Calculation:

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