Q. 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) $\frac{\log_{2}e}{x\log_{e}x}$
Explanation
(d)
Given, $y = \log_{2}\log_{2}{(x)}$
$= \frac{\log_{e}{\log_{2}{(x)}}}{\log_{e}2} = \frac{\log_{e}\left\lbrack \frac{\log_{e}x}{\log_{e}2} \right\rbrack}{\log_{e}2}$
$\Rightarrow y = \frac{\log_{e}{\log_{e}{x - \log_{e}{\log_{e}2}}}}{\log_{e}2}$
On differentiating w.r.t.$x$, we get
$\frac{dy}{dx} = \frac{1}{\log_{e}2}\left\lbrack \frac{1}{x\log_{e}x} - 0 \right\rbrack = \frac{\log_{2}e}{x\log_{e}x}\$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.