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} $