Question

Find the derivative of $\frac{x^2\cos \frac{\pi }{4}}{\sin x}.$

Answer 1

Let $f\left(x\right)=\frac{x^2\cos \frac{\pi }{4}}{\sin x}$
$\Rightarrow f\left(x\right)=\frac{x^2}{\sqrt{2}\sin x}[\because \cos \frac{\pi }{4}=\frac{1}{\sqrt{2}}]$
Differentiating with respect to $x$
$\Rightarrow f^{\prime }\left(x\right)=\frac{d}{dx}\left(\frac{x^2}{\sqrt{2}\sin x}\right)$
$\Rightarrow f^{\prime }\left(x\right)=\frac{1}{\sqrt{2}}\frac{d}{dx}\left(\frac{x^2}{\sin x}\right)$

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

$\Rightarrow f^{\prime }\left(x\right)=\frac{2x\left(\sin x\right)-\left(x^2\right)\left(\cos x\right)}{\sqrt{2}{\sin }^{2}x}$
$\Rightarrow f^{\prime }\left(x\right)=\frac{x(2\sin x-x\cos x)}{\sqrt{2}{\sin }^{2}x}$