Chapter 4.3, Problem 21E

To determine

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

Answer to Problem 21E

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

Explanation of Solution

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

  $y={\tan }^{-1}\left(\sqrt{\left(x^2-1\right)}\right)+{\csc }^{-1}\left(x\right)$

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

Calculation: Using the derivation,

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

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