Question

If $f(x)=\log _x\left(\log _e x\right)$, then $f^{\prime}(x)$ at $x=e$ is equal to

(A) 1

(B) 2

(C) 0

(D) $\frac{1}{e}$

Answer 1

$f(x)=\log _x\left(\log _e x\right)=\frac{\log _e \log _e x}{\log _e x}$

$\begin{aligned}\Rightarrow \quad f^{\prime}(x) & =\frac{\log _e x \cdot \frac{1}{\log _e x} \cdot \frac{1}{x}-\log _e \log _e x \cdot \frac{1}{x}}{\left(\log _e x\right)^2} \\\Rightarrow \quad f^{\prime}(x) & =\frac{1-\log _e \log _e x}{x\left(\log _e x\right)^2} \\\Rightarrow \quad f^{\prime}(e) & =\frac{1-\log _e \log _e e}{e\left(\log _e e\right)^2} \\& =\frac{1-\log _e 1}{e}=\frac{1}{e}\end{aligned}$

So, The correct option will be (D).