Question

Differentiate ${\left(\log x\right)}^{\cos x}$ w.r.t $x$.

Answer 1

$y={\left(\log x\right)}^{\cos x}$
$\log y=\log {\left[\left(\log x\right)\right]}^{\cos x}$
$\log y=\cos x\log \left(\log x\right)$
Differentiating w.r.t $x$
$\frac{1}{y}\frac{dy}{dx}=\cos x\frac{1}{\log x}\frac{1}{x}-\log \left(\log x\right)\sin x$
$\frac{dy}{dx}=y[\frac{\cos x}{x\log x}-\sin \log \left(\log x\right)]$
$\frac{dy}{dx}={\left(\log x\right)}^{\cos x}[\frac{\cos x}{x\log x}-\sin \log \left(\log x\right)]$