Question

Differentiate the following w.r.t. $x$:$(log\,x)^{cos\,x}$

Answer 1

The correct answer is $\frac{dy}{dx}=(log\,x)^{cos\,x}[\frac{cos\,x}{xlog\,x}-sin\,xlog(log\,x)]$
Let $y=(log\,x)^{cos\,x}$
Taking logarithm on both the sides,we obtain
$log\,y=cos\,x.log(log\,x)$
Differentiating both sides with respect to x,we obtain
$\frac{1}{y}.\frac{dy}{dx}=\frac{d}{dx}(cos\,x)\times log(log\,x)+cos\,x\times \frac{d}{dx}[log(log\,x)]$
$⇒\frac{1}{y}.\frac{dy}{dx}=-sin\,xlog(log\,x)+cos\,x\times\frac{1}{log\,x}.\frac{d}{dx}(log\,x)$
$⇒\frac{dy}{dx}=y[-sin\,xlog(log\,x)+\frac{cos\,x}{log\,x}.\frac{1}{x}]$
$∴\frac{dy}{dx}=(log\,x)^{cos\,x}[\frac{cos\,x}{xlog\,x}-sin\,xlog(log\,x)]$