Question

Differentiate the function w.r.t.x. : $(\tan x)^{\sin x}$

Answer 1

we have to differentiate , $y=(tanx)^{(sinx)}$

first of all, take log both sides,

$logy=log\left[(tanx)^{(sinx)}\right]\\\\logy=(sinx)log(tanx)$

differentiate both sides,

$\frac{d(logy)}{dx}=\frac{d\{(sinx)log(tanx)\}}{dx}\\\\\frac{1}{y}\frac{dy}{dx}=(sinx)\frac{d\{log(tanx)\}}{dx}+log(tanx)\frac{d(sinx)}{dx}\\\\\frac{dy}{dx}=y\left[(sinx)\frac{1}{tanx}\frac{d(tanx)}{dx}+log(tanx)cosx\right]\\\\\frac{dy}{dx}=(tanx)^{(sinx)}\left[(sinx)\frac{sec^2x}{tanx}+(cosx)log(tanx)\right]\\\\\frac{dy}{dx}=(tanx)^{(sinx)}[secx+(cosx)log(tanx)]$