Chapter 4.3, Problem 2E

To determine

To calculate: The derivative of y with respect to x .

Answer to Problem 2E

  $\frac{dy}{dx}=\frac{1}{x\times \sqrt{\left(x^2-1\right)}}$

Explanation of Solution

Given information: The equation of y in terms of x is given as,

  $y={\cos }^{-1}\left(\frac{1}{x}\right)$

Formula Used: $\frac{d\left(x^n\right)}{dx}=n\times x^{\left(n-1\right)}$

Calculation: Using the derivation,

  $\begin{array}{l}\frac{dy}{dx}=\frac{d\left({\cos }^{-1}\left(\frac{1}{x}\right)\right)}{dx}\\ \frac{dy}{dx}=\frac{-1}{\sqrt{\left(1-{\left(\frac{1}{x}\right)}^2\right)}}\times \frac{d\left(\frac{1}{x}\right)}{dx}\\ \frac{dy}{dx}=\frac{-1}{\sqrt{\left(1-\frac{1}{x^2}\right)}}\times \frac{-1}{x^2}\\ \frac{dy}{dx}=\frac{x}{x^2\times \sqrt{\left(x^2-1\right)}}\\ \frac{dy}{dx}=\frac{1}{x\times \sqrt{\left(x^2-1\right)}}\end{array}$

So the derivative of y with respect to x is $\frac{dy}{dx}=\frac{1}{x\times \sqrt{\left(x^2-1\right)}}$ .