If `f(x)=(log)_(x^2)(logx),` then `f^(prime)(x)` at `x=e` is 0 (b) 1 (c) `1/e` (d) `1/2e`

Explanation

To find the derivative \( f'(x) \) of the function \( f(x) = \log_{x^2}(\log x) \) and evaluate it at \( x = e \), we will follow these steps:

### Step 1: Rewrite the function using logarithm properties
The function can be rewritten using the change of base formula for logarithms:
\[ f(x) = \frac{\log(\log x)}{\log(x^2)} \]
Using the property \( \log(x^2) = 2\log x \), we can simplify this to:
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