The differentiation of e^tan^-1x is equal to 1/(1+x^{2}) e^{tan^{-1}x}. 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^{-1}x} with respect to x.

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

^{-1}x}

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

**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(e^{k}) =k. Therefore, we obtain that

ln y = tan^{-1}x

**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^{-1}x is 1/(1+x^{2}).

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

**Third step:**

Put the value of y, i.e., y=e^{tan^{-1}x}. 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 e ^{tan-1x}/(1+x^{2}).**

**ALSO READ:**

Differentiate e^{sinx}: The derivative of e^{sinx} is e^{sinx} cosx.

Differentiate e^{cosx}: The derivative of e^{cosx} is -e^{cosx} sinx.

Differentiate e^{tanx}: The derivative of e^{tanx} is e^{tanx} sec^{2}x.

Differentiate e^{sin^{-1}x}: The derivative of e to the power sin inverse x is e^{sin-1x}/√(1-x^{2}).

Differentiate e^{cos^{-1}x}: The derivative of e to the power cos inverse x is -e^{cos-1x}/√(1-x^{2}).

## FAQs

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

Answer: The derivative of e to the power tan^{-1}x is equal to e^{tan-1x}/(1+x^{2}).

### Q2: If y=e^{tan-1x}, then find dy/dx?

Answer: If y=e^{tan-1x}, then dy/dx = e^{tan-1x}/(1+x^{2}).