Question

Differentiate $\frac{e^x-tanx}{cotx-x^n}$ w.r.t $x$.

Answer 1

$y=\frac{e^x-\tan x}{\cot x-x^x}$

$\frac{dy}{dx}=\frac{(\cot x-x^n)\frac{d}{dx}(e^x-\tan x)-(e^x-\tan x)\frac{d}{dx}(\cot x-x^n)}{{(\cot x-x^x)}^2}$

$=\frac{(\cot x-x^x)(e^x-{\sec }^{2}x)-(e^x-\tan x)(-{\csc }^{2}x-n{x}^{n-1})}{{(\cot x-x^x)}^2}$

$=\frac{e^x\cot x-x^n{e}^x-\cot x{\sec }^{2}x+x^n{\sec }^{2}x+e^x{\csc }^{2}x+n{e}^x{x}^{n-1}-\tan x{\csc }^{2}x-n{x}^{n-1}\tan x}{{(\cot x-x^x)}^2}$