Chapter A6, Problem 35E

To determine

To find: the derivative of y

Answer to Problem 35E

The derivative of y is $\Rightarrow \left|\frac{dy}{dx}=\right|\sec x$

Explanation of Solution

Given:

  $y={\sinh }^{-1}(\tan x)$

Calculation:

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

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

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

  $\Rightarrow \frac{dy}{dx}=\frac{{\sec }^{2}x}{\left|\sec x\right|}$

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

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

Conclusion:

Therefore,The derivative of y is $\Rightarrow \left|\frac{dy}{dx}=\right|\sec x$