Question

If $y = \log (\log x)$ then the value of $e^y \frac{dy}{dx} $ is

• A. $e^y $
• B. $\frac{1}{x}$
• C. $\frac{1}{(\log \, x)}$
• D. $\frac{1}{(x \, \log \, x)}$

Answer 1

Answer: (B)

Let $y = \log (\log x)$
Diff both side w.r.t $'x'$, we get
$\frac{dy}{dx} = \frac{1}{log \,x } \frac{d}{dx} (\log \, x)$
$\Rightarrow \:\: \frac{dy}{dx} = \frac{1}{\log x } \times \frac{1}{x}$
$\Rightarrow \:\: \log \, x .\frac{dy}{dx} = \frac{1}{x}$
$\Rightarrow \:\: e^y \frac{dy}{dx} = \frac{1}{x} \:\:\:$
$ (\because \:\: e^y = e^{\log(\log \,x)} = \log \, x)$