Question

find derivative of cosec x using product rule.

Answer 1

$\begin{array}{l}\text{We can write}\cos \text{ec}x\text{as}(\sec x.\cot x)\\ \text{Now,}\frac{d}{dx}(\sec x.\cot x)\\ =\sec x\frac{d}{dx}\left(\cot x\right)+\cot x\frac{d}{dx}\left(\sec x\right)\\ =\sec x(-\cos {\text{ec}}^{2}x)+\cot x\left(\sec x\tan x\right)\\ =\sec x(-\cos {\text{ec}}^{2}x+\cot x\tan x)\\ =\sec x(-{\text{cosec}}^{2}x+1)\\ =\frac{1}{\cos x}.(-{\cot }^{2}x)\\ =\frac{1}{\cos x}.\left(\frac{-{\cos }^{2}x}{\sin x}\right)\\ =-\frac{\cos x}{\sin x}.\frac{1}{\sin x}\\ =-\cot x.\mathrm{cosec}x.\end{array}$