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

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

Formula used: $\frac{\mathrm{d}}{\mathrm{dx}}(\sqrt{\text { usinu }})=\frac{1}{2 \sqrt{\text { usinu }}} \times \frac{\mathrm{d}}{\mathrm{dx}}($ usinu $)$

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

So, by using the above formula, we have

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

Differentiation of $y=\sqrt{x \sin x}$ is $\frac{(\sin x+x \cos x)}{2 \sqrt{x \sin x}}$