Question

If $y={\log }_2{\log }_2(x),$ then $\frac{dy}{dx}$ is equal to

• A. $\frac{{\log }_{2}e}{{\log }_{e}x}$
• B. $\frac{{\log }_{2}e}{x{\log }_{x}2}$
• C. $\frac{{\log }_{2}x}{{\log }_{e}2}$
• D. $\frac{{\log }_{2}e}{{\log }_{2}x}$
• E. $\frac{{\log }_{2}e}{x{\log }_{e}x}$

Answer 1

Answer: (E)

Given, $y={\log }_2{\log }_2(x)$
$=\frac{{\log }_e{\log }_2(x)}{{\log }_{e}2}$
$=\frac{{\log }_e\left[\frac{{\log }_{e}x}{{\log }_{e}2}\right]}{{\log }_{e}2}$
$\Rightarrow$ $y=\frac{{\log }_e{\log }_{e}x-{\log }_e{\log }_{e}2}{{\log }_{e}2}$
On differentiating w.r.t. $x,$ we get
$\frac{dy}{dx}=\frac{1}{{\log }_{e}2}\left[\frac{1}{x{\log }_{e}x}-0\right]$
$=\frac{{\log }_{2}e}{x{\log }_{e}x}$