Question

Find the derivative of the following function:
f(x) = $\frac{x}{si{n}^{n}x}$

Answer 1

Here $f\left(x\right)=\frac{x}{si{n}^{n}x}$

$\therefore f^{\prime }\left(x\right)=\frac{d}{dx}\left[\frac{x}{si{n}^{n}x}\right]$

$=\frac{si{n}^{n}x\frac{d}{dx}\left(x\right)-x\frac{d}{dx}\left(si{n}^{n}x\right)}{(si{n}^{n}x{)}^2}$

$=\frac{si{n}^{n}x\times 1-x.nsi{n}^{n-1}x\frac{d}{dx}\left(sinx\right)}{(si{n}^{n}x{)}^2}$

$=\frac{si{n}^{n}x-nxsi{n}^{n-1}xcosx}{(si{n}^{n}x{)}^2}$

$=\frac{si{n}^{n-1}x[sinx-nxcosx]}{si{n}^{2n}x}$

$=\frac{sinx-nxcosx}{si{n}^{n+1}x}$