Question:
Choose the correct alternative in the following:
The differential coefficient of $f(\log x)$ with respect to $x$, where $f(x)=\log x$ is
A. $\frac{\mathrm{x}}{\log \mathrm{x}}$
B. $\frac{\log x}{x}$
C. $(x \log x)^{-1}$
D. none of these
Solution:
Given: $f(x)=\log x$
$\therefore f(\log x)=\log (\log x)$
$f^{\prime}(\log x)=\frac{d}{d x} \log (\log x)$
$f^{\prime}(\log x)=\frac{1}{\log x} \cdot \frac{1}{x}=\frac{1}{x \log x}$
$\therefore f^{\prime}(\log x)=(x \log x)^{-1}$