Question

Differentiate the function given below w.r.t. $x:$

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

Answer 1

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

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

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


$\Rightarrow \frac{dy}{dx}=\frac{1}{\sqrt{2}}\left[\frac{x[2\sin x-x\cos x]}{(\sin x{)}^2}\right]$

$\text{divide by}(\sin x{)}^2$

$\Rightarrow \frac{dy}{dx}=\frac{x}{\sqrt{2}}[2\csc x-x\csc x\cot x]$

$\Rightarrow \frac{dy}{dx}=\frac{x}{\sqrt{2}}\csc x[2-x\cot x]$