If $ y=\log (\log x) $ then $ \frac{d^{2} y}{d x^{2}} $ is equal to

Question

If $ y=\log (\log x) $ then $ \frac{d^{2} y}{d x^{2}} $ is equal to, $ -\frac{(1+\log x)}{(x \log x)^{2}} $, $ -\frac{(1+\log x)}{x^{2} \log x} $, $ \frac{(1+\log x)}{(x \log x)^{2}} $, $ \frac{(1+\log x)}{x^{2} \log x} $

MARKS App · Accepted Answer

Given that, $ y=\log (\log x) \rightarrow(1) $ Differentiate Eq. (1) with respect to $ x $, we get $$ \frac{d y}{d x}=\frac{1}{\log x}\left(\frac{1}{x}\right)=\frac{1}{x \log x} $$ Again, differentiate Eq. (2) with respect to $ x $, we get $$ \begin{array}{l} \frac{d^{2} y}{d x}=\frac{-1}{(x \log x)^{2}} \cdot\left[x \cdot \frac{1}{x}+\log x \cdot 1\right] \ =\frac{-(1+\log x)}{(x \log x)^{2}} \end{array} $$

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