Question

Find the derivative :$\cot x$

Answer 1

Given,

$\frac{d}{dx}\cot x=\frac{d}{dx}\left(\frac{\cos x}{\sin x}\right)=\frac{\sin x\frac{d}{dx}\cos x-\cos x\frac{d}{dx}\sin x}{{\sin }^{2}x}=\frac{\sin x(-\sin x)-\cos x\left(\cos x\right)}{{\sin }^{2}x}=\frac{-{\sin }^{2}x-{\cos }^{2}x}{{\sin }^{2}x}=-\left(\frac{{\sin }^{2}x+{\cos }^{2}x}{{\sin }^{2}x}\right)=-\frac{1}{{\sin }^{2}x}=-cose{c}^{2}x.$