Question

Both of the following equations are true: $\tan \left({\tan }^{-1}x\right)=x\text{and}{\tan }^{-1}\left(\tan x\right)=x$. We can find the derivative of ${\tan }^{-1}x$by differentiating both sides of the equations and solving for$\frac{d}{dx}{\tan }^{-1}x$.Which one of the equations should we use, and why ?

Short answer

We will use the equation $\tan \left({\tan }^{-1}x\right)=x$as it is given that both sides can be differentiated.

Step-by-step solution

Step 1. Given information.

It is given that

$$\begin{gathered} \tan \left({\tan }^{-1}x\right)=x \\ {\tan }^{-1}\left(\tan x\right)=x \end{gathered}$$

Step 2. Explanation.

We should use the equation${\tan }^{-1}\left(\tan x\right)=x$ , because here it is given that the derivative of ${\tan }^{-1}x$ can be find out by differentiating both sides of the equation.