Chapter A6, Problem 36E

To determine

To find: the derivative of y

Answer to Problem 36E

The derivative of y is $\Rightarrow \frac{dy}{dx}=\sec x,0<x<\frac{\pi }{2}$

Explanation of Solution

Given:

  $y={\cosh }^{-1}(\sec x)$

Calculation:

Given, $y={\cosh }^{-1}(\sec x)$

To find $\frac{dy}{dx},$ differentiate $y$ with respect to $x$

  $\Rightarrow \frac{dy}{dx}=\frac{(\sec x)(\tan x)}{\sqrt{{(\sec x)}^2-1}}$

  $\Rightarrow \frac{dy}{dx}=\frac{(\sec x)(\tan x)}{\sqrt{{\tan }^{2}x}}$

  $\Rightarrow \frac{dy}{dx}=\frac{(\sec x)(\tan x)}{\left|\tan x\right|}$

  $\Rightarrow \frac{dy}{dx}=\sec x,0<x<\frac{\pi }{2}$

Conclusion:

Therefore, The derivative of y is $\Rightarrow \frac{dy}{dx}=\sec x,0<x<\frac{\pi }{2}$