Question

Find the derivatives w.r.t.x:
$\rm \frac{e^{x}-\tan x}{\cot x-x^{n}}$

Answer 1

here is ur Answer ✍️✍️

Answer 2

We know that such form of expression can solve by U/V method of differentiation

$\begin{lgathered}\frac{d}{dx} \bigg(\frac{U}{V}\bigg ) = \frac{V \frac{dU}{dx} - U \frac{dV}{dx} }{ {V}^{2} } \\ \\ here \: U = e^{x}-\tan x\\ \\ V = \cot x-x^{n} \\\\\end{lgathered}$

$\frac{d}{dx} \bigg(\frac{e^{x}-\tan x}{\cot x-x^{n}}\bigg) \\ \\ = \frac{(\cot x-x^{n}) \frac{d(e^{x}-\tan x)}{dx} - (e^{x}-\tan x )\frac{d(\cot x-x^{n})}{dx} }{( \cot x-x^{n})^{2} } \\ \\ = \frac{(\cot x-x^{n}) (e^{x}-sec^{2} x)- (e^{x}-\tan x )(-\cosec^2{x}-nx^{(n-1)}) }{( \cot x-x^{n})^{2} } \\ \\ \frac{d}{dx} \bigg(\frac{e^{x}-\tan x}{\cot x-x^{n}}\bigg)\\\\= \frac{(\cot x-x^{n}) (e^{x}-sec^{2} x)+ (e^{x}-\tan x )(\cosec^2{x}+nx^{(n-1)}) }{( \cot x-x^{n})^{2} }$
Hope it helps you.