Question

Differentiate the following functions with respect to x :

$\frac{x}{si{n}^{n}x}$

Answer 1

Let $y=\frac{x}{si{n}^{n}x}$

Differentiating y w.r.t. x, we get

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

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

Differentiating $(sinx{)}^{n-1}$ by chain rule,

$=\frac{si{n}^{n}x-nx\left(sinx{)}^{n-1}\frac{d}{dx}\right(sinx)}{si{n}^{2n}x}=\frac{si{n}^{n}x-nxsi{n}^{n-1}xcosx}{si{n}^{2n}x}$