Question

Differentiate the following functions with respect to x :

$\frac{xsinx}{1+cosx}$

Answer 1

We have,

$\frac{d}{dx}\left(\frac{xsinx}{1+cosx}\right)$

Using quotient rule, we get

$=\frac{(1+cosx)\frac{d}{dx}\left(xsinx\right)-\left(xsinx\right)\frac{d}{dx}(1+cosx)}{(1+cosx{)}^2}$

$=\frac{(1+cosx)(x\frac{d}{dx}sinx+sinx\frac{d}{dx})-xsinx-(-sinx)}{(1+cosx{)}^2}$

$=\frac{(1+cosx)(xcosx+sinx)+xsi{n}^{2}x}{(1+cosx{)}^2}$

$=\frac{xcosx+xco{s}^{2}x+sinx+sinxcosx+xsi{n}^{2}x}{(1+cosx{)}^2}$

$=\frac{(xcosx+sinx+sinxcosx)+x(si{n}^{2}x+co{s}^{2}x)}{(1+cosx{)}^2}=\frac{sinx+sincosx+x(cosx+si{n}^{2}x+co{s}^{2}x)}{(1+cosx{)}^2}$

$=\frac{sinx(1+cosx)+x(cosx+1)}{(1+cosx{)}^2}=\frac{(x+sinx)(cosx+1)}{(1+cosx{)}^2}$