If \(f(x)=\log _{x^{2}}\left(\log _{e} x\right)\), then \(f^{\prime}(x)\) at \(x=e\) is

Given, \(\mathrm{f}(\mathrm{x})=\log _{\mathrm{x}^{2}}\left(\log _{\mathrm{e}} \mathrm{x}\right)=\frac{1}{2} \log _{\mathrm{x}}\left(\log _{\mathrm{e}} \mathrm{x}\right)\)
\(\Rightarrow \quad \mathrm{f}(\mathrm{x})=\frac{1}{2} \frac{\log _{\mathrm{e}} \log _{\mathrm{e}} \mathrm{x}}{\log _{\mathrm{e}} \mathrm{x}}\)
\(\Rightarrow \mathrm{f}^{\prime}(\mathrm{x})=\frac{1}{2} \frac{\log _{\mathrm{e}} \mathrm{x}\left[\frac{1}{\mathrm{x} \log _{\mathrm{e}} \mathrm{x}}\right]-\log _{\mathrm{e}} \log _{\mathrm{e}} \mathrm{x} \times \frac{1}{\mathrm{x}}}{\left(\log _{\mathrm{e}} \mathrm{x}\right)^{2}}\)
\(\Rightarrow \quad \mathrm{f}^{\prime}(\mathrm{x})=\frac{1}{2} \frac{\frac{1}{\mathrm{x}}-\frac{1}{\mathrm{x}} \log _{\mathrm{e}} \log _{\mathrm{e}} \mathrm{x}}{\left(\log _{\mathrm{e}} \mathrm{x}\right)^{2}}\)
At \(\mathrm{x}=\mathrm{e}, \mathrm{f}^{\prime}(\mathrm{e})=\frac{1}{2} \frac{\frac{1}{\mathrm{e}}-\frac{1}{\mathrm{e}} \log _{\mathrm{e}} 1}{(1)^{2}}\)
\(\Rightarrow \quad \mathrm{f}^{\prime}(\mathrm{e})=\frac{1}{2 \mathrm{e}}\)