Question

If $y=\log \left[\log \right(\log x\left)\right]$, then value of $\frac{dy}{dx}$ is.

• A. $\frac{1}{x\log x}$
• B. $\frac{1}{x\log x\log \left(\log x\right)}$
• C. $\frac{1}{x\log \left(\log x\right)}$
• D. $\frac{1}{x}$

Answer 1

The correct option is B $\frac{1}{x\log x\log \left(\log x\right)}$

$y=\log \left[\log \right(\log x\left)\right]$

Using $\frac{d\left(fog\right(x\left)\right)}{dx}=\frac{d\left(f\right(g\left(x\right)\left)\right)}{dx}\times \frac{d\left(g\right(x\left)\right)}{dx}$, we have

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

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

$\Rightarrow \frac{dy}{dx}=\frac{1}{\log \left(\log x\right)}\times \frac{1}{\log x}\times \frac{1}{x}$