# Differentiation of e^tan^-1x

The differentiation of e^tan^-1x is equal to 1/(1+x2) e^{tan-1x}. In this post, we will learn how to differentiate e to power tan inverse x with respect to x.

We will now find the derivative of e^{tan-1x} with respect to x.

## Find the derivative of e^{tan-1x}

Question: How to differentiate e^{tan-1x}?

Solution:

We will follow the below steps in order to get the derivative of e to the power tan inverse x. The first step is as follows.

First step:

Let y= $e^{\tan^{-1}x}$

We take natural logarithms on both sides and use the logarithm formula ln(ek) =k. Therefore, we obtain that

ln y = tan-1x

Second step:

Differentiating both sides with respect to x, we get that

$\dfrac{1}{y} \dfrac{dy}{dx}$ = $\dfrac{1}{1+x^2}$ as the derivative of tan-1x is 1/(1+x2).

⇒ $\dfrac{dy}{dx}$ = $\dfrac{y}{1+x^2}$

Third step:

Put the value of y, i.e., y=e^{tan-1x}. This will imply that

$\dfrac{d}{dx} \Big(e^{\tan^{-1}x} \Big)$ = $\dfrac{e^{\tan^{-1}x}}{1+x^2}$.

Thus, the differentiation of e to the power tan inverse x with respect to x is equal to etan-1x/(1+x2).

Differentiate esinx: The derivative of esinx is esinx cosx.

Differentiate ecosx: The derivative of ecosx is -ecosx sinx.

Differentiate etanx: The derivative of etanx is etanx sec2x.

Differentiate e^{sin-1x}: The derivative of e to the power sin inverse x is esin-1x/√(1-x2).

Differentiate e^{cos-1x}: The derivative of e to the power cos inverse x is -ecos-1x/√(1-x2).

## FAQs

### Q1: What is the derivative of e to the power tan inverse x?

Answer: The derivative of e to the power tan-1x is equal to etan-1x/(1+x2).

### Q2: If y=etan-1x, then find dy/dx?

Answer: If y=etan-1x, then dy/dx = etan-1x/(1+x2).