Question

If f (x) = log_x² (log x), the f' (x) at x = e is
(a) 0
(b) 1
(c) 1/e
(d) 1/2 e

Answer 1

(d) 1/2 e

$$\begin{gathered} \mathrm{We}\mathrm{have},f\left(x\right)={\log }_{x^2}\left(\log x\right) \\ \Rightarrow f\left(x\right)=\frac{\log \left(\log x\right)}{\log x^2} \\ \Rightarrow f\left(x\right)=\frac{\log \left(\log x\right)}{2\log x} \\ \Rightarrow f\text{'}\left(x\right)=\frac{1}{2}\times \frac{d}{dx}\left\{\frac{\log \left(\log x\right)}{\log x}\right\} \\ \Rightarrow f\text{'}\left(x\right)=\frac{1}{2}\times \left\{\frac{{\displaystyle \frac{1}{\log x}}\times {\displaystyle \frac{1}{x}}\times \log x-{\displaystyle \frac{\log \left(\log x\right)}{x}}}{{\left(\log x\right)}^2}\right\} \\ \Rightarrow f\text{'}\left(x\right)=\frac{1}{2}\times \left\{\frac{{\displaystyle \frac{1}{x}}-{\displaystyle \frac{\log \left(\log x\right)}{x}}}{{\left(\log x\right)}^2}\right\} \\ \Rightarrow f\text{'}\left(e\right)=\frac{1}{2}\times \left\{\frac{{\displaystyle \frac{1}{e}}-{\displaystyle \frac{\log \left(\log e\right)}{e}}}{{\left(\log e\right)}^2}\right\}\left[\mathrm{Putting}x=e\right] \\ \Rightarrow f\text{'}\left(e\right)=\frac{1}{2}\times \left\{\frac{{\displaystyle \frac{1}{e}}}{1}\right\} \\ \Rightarrow f\text{'}\left(e\right)=\frac{1}{2e} \end{gathered}$$