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

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

Formula used: $\frac{d}{d x}(\sqrt{\sin n u})=\frac{1}{2 \sqrt{\operatorname{sinn} u}} \times \frac{d}{d x}(\operatorname{sinnu}) \times \frac{d}{d x}(n u)$ and $\frac{d x^{n}}{d x}=n x^{n-1}$

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

So, by using the above formula, we have

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

Differentiation of $y=\sqrt{\sin x}$ is $\frac{1}{2 \sqrt{\sin x}} \cos x$