Differentiate the following functions with respect to x :

Let $y=3^{\mathrm{e}^{\mathrm{x}}}$

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

$\frac{d y}{d x}=\frac{d}{d x}\left(3^{e^{x}}\right)$

We know $\frac{d}{d x}\left(a^{x}\right)=a^{x} \log a$

$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=3^{\mathrm{e}^{\mathrm{x}}} \log 3 \frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{e}^{\mathrm{x}}\right)$ [using chain rule]

We have $\frac{d}{d x}\left(e^{x}\right)=e^{x}$

$\Rightarrow \frac{d y}{d x}=3^{e^{x}} \log 3 \times e^{x}$

$\therefore \frac{\mathrm{dy}}{\mathrm{dx}}=3^{\mathrm{e}^{\mathrm{x}}} \mathrm{e}^{\mathrm{x}} \log 3$

Thus, $\frac{\mathrm{d}}{\mathrm{dx}}\left(3^{\mathrm{e}^{\mathrm{x}}}\right)=3^{\mathrm{e}^{\mathrm{x}}} \mathrm{e}^{\mathrm{x}} \log 3$