Question

Differentiate the following questions w.r.t. x.

$\frac{cosx}{logx}$, x>0.

Answer 1

Let y = $\frac{cosx}{logx}$

Differentiate both sides w.r.t. x,

$\Rightarrow \frac{dy}{dx}=\frac{d}{dx}\left(\frac{cosx}{logx}\right),x>0\Rightarrow \frac{dy}{dx}=\frac{\left(logx\right)\frac{dy}{dx}\left(cosx\right)-cosx\frac{d}{dx}\left(logx\right)}{(logx{)}^2}(\because \frac{d}{dx}\left(\frac{u}{v}\right)=\frac{v\frac{d}{dx}\left(u\right)-u\frac{d}{dx}\left(v\right)}{v^2})\Rightarrow \frac{dy}{dx}=\frac{\left(logx\right)(-sinx)-\left(cosx\right)\frac{1}{x}}{(logx{)}^2}\Rightarrow \frac{dy}{dx}=-\frac{xsinxlogx+cosx}{x(logx{)}^2},x>0$