Question

Differentiate $\log \left(\log x\right),x>1$ w.r.t. $x$

Answer 1

Let $y=\log \left(\log x\right),x>1$

Differentiating both side w.r.t. $x$, we get,

$\frac{d\left(y\right)}{dx}=\frac{d\left(\log \right(\log x\left)\right)}{dx}$

$\frac{d\left(y\right)}{dx}=\frac{1}{\log x}\times \frac{d\left(\log x\right)}{dx}$

$(\text{using chain rule}\frac{dy}{dx}=\frac{dy}{du}\times \frac{du}{dx})$

$\frac{dy}{dx}=\frac{1}{\log x}\times \frac{1}{x}$

$\frac{dy}{dx}=\frac{1}{x\log x}$