Question

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

Answer 1

Let $y=(\log x{)}^{\cos x}$
Taking logarithm on both the sides, we obtain
$\log y=\cos x.\log \left(\log x\right)$ ..... $[\because \log a^x=x\log a]$
Differentiating both sides with respect to $x$, we obtain
$\frac{1}{y}.\frac{dy}{dx}=\frac{d}{dx}\left(\cos x\right)\times \log \left(\log x\right)+\cos x\times \frac{d}{dx}\left[\log \right(\log x\left)\right]$
$\Rightarrow \frac{1}{y}.\frac{dy}{dx}=-\sin x\log \left(\log x\right)+\cos x\times \frac{1}{\log x}.\frac{d}{dx}\left(\log x\right)$
$\Rightarrow \frac{dy}{dx}=y[-\sin x\log (\log x)+\frac{\cos x}{\log x}\times \frac{1}{x}]$
$\therefore \frac{dy}{dx}=(\log x{)}^{\cos x}[\frac{\cos x}{x\log x}-\sin x\log (\log x\left)\right]$