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

Explanation

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

### Step 1: Rewrite the function using the change of base formula
The change of base formula for logarithms states that:
\[ \log_a(b) = \frac{\log_c(b)}{\log_c(a)} \]
Using this, we can rewrite \( f(x) \):
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