Differentiate the following functions with respect to x: ^ -1 (e^… - Vidyadip

Question

Differentiate the following functions with respect to x:
$\tan^{-1}(\text{e}^{\text{x}})$

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Answer

Consider $\text{y}=\tan^{-1}(\text{e}^{\text{x}})$
Differentiate with respect to x,
$\frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}\big(\tan^{-1}\text{e}^{\text{x}}\big)$
$=\frac{1}{1+\big(\text{e}^{2\text{x}}\big)^2}=\frac{\text{d}}{\text{dx}}\big(\text{e}^\text{x}\big)$
[using chain rule]
$=\frac{1}{1+\text{e}^{2\text{x}}}\times\text{e}^\text{x}$
$=\frac{\text{e}^\text{x}}{1+\text{e}^{2\text{x}}}$
Hence, the solution is, $\frac{\text{d}}{\text{dx}}\big(\tan^{-1}\text{e}^\text{x}\big)=\frac{\text{e}^\text{x}}{1+\text{e}^{2\text{x}}}$

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