Differentiate the following with respect to x:

To Find: Differentiation

NOTE : When 2 functions are in the product then we used product rule i.e

$\frac{\mathrm{d}(\mathrm{u} \cdot \mathrm{v})}{\mathrm{dx}}=\mathrm{V} \frac{\mathrm{du}}{\mathrm{dx}}+\mathrm{u} \frac{\mathrm{dv}}{\mathrm{dx}}$c

Formula used: $\frac{d}{d x}\left(\sqrt{\sin u^{a}}\right)=\frac{1}{2 \sqrt{\sin u^{a}}} \times \frac{d}{d x}\left(\sin u^{a}\right) \times \frac{d}{d x}\left(u^{a}\right)$

Let us take $y=\sqrt{\sin x^{3}}$

So, by using the above formula, we have

$\frac{d}{d x} \sqrt{\sin x^{3}}=\frac{1}{2 \sqrt{\sin x^{3}}} \times \frac{d}{d x}\left(\sin x^{3}\right) \times \frac{d}{d x}\left(x^{3}\right)=\frac{1}{2 \sqrt{\sin x^{3}}} \times\left(\cos x^{3}\right) \times 3 x^{2}=\frac{3 x^{2}\left(\cos x^{3}\right)}{2 \sqrt{\sin x^{3}}}$

Differentiation of $y=\sqrt{\sin x^{3}}$ is $\frac{3 x^{2}\left(\cos x^{3}\right)}{2 \sqrt{\sin x^{3}}}$