Question

The differential coefficient of f (log x) w.r.t. x, where f (x) = log x is
(a) $\frac{x}{\log x}$

(b) $\frac{\log x}{x}$

(c) ${\left(x\log x\right)}^{-1}$

(d) none of these

Answer 1

(c) ${\left(x\log x\right)}^{-1}$

We have,
$$\begin{gathered} f\left(x\right)=\log x \\ \Rightarrow f\left(\log x\right)=\log \left(\log x\right) \\ \Rightarrow f\text{'}\left(\log x\right)=\frac{1}{\log x}\frac{d}{dx}\left(\log x\right) \\ \Rightarrow f\text{'}\left(\log x\right)=\frac{1}{x\log x} \\ \Rightarrow f\text{'}\left(\log x\right)={\left(x\log x\right)}^{-1} \end{gathered}$$