# Differentiate the following w.r.t $x$ : $log\begin{bmatrix}log\bigg(log x^5\bigg)\end{bmatrix}$.

$\begin{array}{1 1}\large\frac{5x}{\log x^5.\log(\log x^5)} \\ \large\frac{5}{x\log x^5.\log(\log x^5)} \\ \large\frac{5\log x^5}{x\log(\log x^5)} \\ \large\frac{x}{\log x^5.\log(\log x^5)}\end{array}$

Toolbox:
• Chain rule : Suppose $f$ is a real valued function which is a composite of three functions $u,v$ and $w$ (i.e) $f=(wou)ov,$ then $\large\frac{df}{dx}=\frac{dw}{ds}$$\times\large\frac{ds}{dt}$$\times\large\frac{dt}{dx}$
Step 1:
$y=\log\mid\log(\log x)\mid$
Let $u=\log x^5$
$\large\frac{du}{dx}=\frac{1}{x^5}$$5x^4=\large\frac{5}{x} Let \log u=v \large\frac{dv}{du}=\large\frac{1}{u} Therefore y=\log v \large\frac{dy}{dv}=\large\frac{1}{v} Therefore \large\frac{dy}{dx}=\frac{dy}{dv}$$\times\large\frac{dv}{du}$$\times\large\frac{du}{dx} \qquad\qquad\;\;\;\;=\large\frac{1}{v}$$\times\large\frac{1}{u}$$\times\large\frac{1}{x^5}$$\times 5x^4$
Step 2:
Substituting for $v$ and $u$ we get,
$\large\frac{dy}{dx}=\frac{1}{\log u}$$\times\large\frac{1}{\log x^5}$$\times \large\frac{5}{x}$
$\quad\;\;=\large\frac{1}{\log (\log x^5)}$$\times\large\frac{1}{\log x^5}$$\times \large\frac{5}{x}$
$\quad\;\;=\large\frac{5}{x\log x^5.\log(\log x^5)}$