Differentiate the following functions with respect to x :

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

On differentiating y with respect to $x$, we get

$\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{e}^{\tan x}\right)$

We know $\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{e}^{\mathrm{x}}\right)=\mathrm{e}^{\mathrm{x}}$

$\Rightarrow \frac{d y}{d x}=e^{\tan x} \frac{d}{d x}(\tan x)$ [using chain rule]

We have $\frac{\mathrm{d}}{\mathrm{dx}}(\tan \mathrm{x})=\sec ^{2} \mathrm{x}$

$\therefore \frac{d y}{d x}=e^{\tan x} \sec ^{2} x$

Thus, $\frac{d}{d x}\left(e^{\tan x}\right)=e^{\tan x} \sec ^{2} x$