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(e^k) =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 e^tan-1x/(1+x²).

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 sec²x.

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

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

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