Differentiate the following functions with respect to x :

Let $y=2^{x^{3}}$

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

$\frac{d y}{d x}=\frac{d}{d x}\left(2^{x^{3}}\right)$

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

$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=2^{\mathrm{x}^{3}} \log 2 \frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{x}^{3}\right)$ [using chain rule]

We have $\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{x}^{\mathrm{n}}\right)=\mathrm{n} \mathrm{x}^{\mathrm{n}-1}$

$\Rightarrow \frac{d y}{d x}=2^{x^{3}} \log 2 \frac{d}{d x}\left(x^{3}\right)$ [using chain rule]

$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=2^{\mathrm{x}^{2}} \log 2 \times 3 \mathrm{x}^{3-1}$

$\Rightarrow \frac{d y}{d x}=2^{x^{3}} \log 2 \times 3 x^{2}$

$\therefore \frac{d y}{d x}=2^{x^{3}} 3 x^{2} \log 2$

Thus, $\frac{\mathrm{d}}{\mathrm{dx}}\left(2^{\mathrm{x}^{3}}\right)=2^{\mathrm{x}^{3}} 3 \mathrm{x}^{2} \log 2$