Question

Differentiate the following function with respect to x.

$\frac{x^2\cos \frac{\pi }{4}}{\sin x}$.

Answer 1

Let $y=\frac{x^2\cos \frac{\pi }{4}}{\sin x}$

Now, differentiate w.r.t. x

$\frac{dy}{dx}=\frac{d}{dx}\left(\frac{x^2\cos \frac{\pi }{4}}{\sin x}\right)$

$=\frac{1}{\sqrt{2}}\frac{d}{dx}\left(x^{2}cosecx\right)$

We know, $\frac{d(u.v)}{dx}=v\frac{du}{dx}+u\frac{dv}{dx}$

$\frac{dy}{dx}=\frac{1}{\sqrt{2}}\left\{cosecx\frac{d}{dx}\right(x^2)+x^2\frac{d}{dx}(cosecx\left)\right\}$

$\frac{dy}{dx}=\frac{1}{\sqrt{2}}(2xcosecx-x^{2}cosecx\cot x)$.