Question

The differential coefficient of $f(\log x)$ where $f\left(x\right)=\log x$, is:

• A. $\frac{x}{\log x}$
• B. ${\left(x\log x\right)}^{-1}$
• C. $\frac{\log x}{x}$
• D. None of these

Answer 1

The correct option is B

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

Differentiating to find the coefficient:

Given that: $f\left(x\right)=\log x$

$\Rightarrow f(\log x)=\log \left(\log x\right)$

Differentiating w.r.t $x$:

$$\begin{gathered} \Rightarrow f\text{'}(\log x)=\frac{1}{\log x}\times \frac{1}{x}\left[\because \frac{d}{dx}\log x=\frac{1}{x}\right] \\ \Rightarrow f\text{'}(\log x)=\frac{1}{x\log x} \\ \Rightarrow f\text{'}(\log x)={\left(x\log x\right)}^{-1} \end{gathered}$$

Hence, the correct answer is Option (B).