If `f(x)=log_(x^(2))(logx)`,then `f '(x) "at" x= e` is

Options

A. 0
B. 1
C. `(1)/(e )`
D. `(1)/(2e)`

Answer

To solve the problem, we need to find the derivative \( f'(x) \) of the function \( f(x) = \log_{x^2}(\log x) \) and then evaluate it at \( x = e \).

### Step-by-Step Solution:

1. **Rewrite the function using logarithmic properties:**
\[ f(x) = \log_{x^2}(\log x) = \frac{\log(\log x)}{\log(x^2)} = \frac{\log(\log x)}{2 \log x} \]

2. **Differentiate \( f(x) \):**
We will use the quotient rule for differentiation, which states that if \( f(x) = \frac{g(x)}{h(x)} \), then:
\[ f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2} \]
Here, \( g(x) = \log(\log x) \) and \( h(x) = 2 \log x \).

3. **Find \( g'(x) \) and \( h'(x) \):**
- For \( g(x) = \log(\log x) \):
\[ g'(x) = \frac{1}{\log x} \cdot \frac{1}{x} = \frac{1}{x \log x} \]
- For \( h(x) = 2 \log x \):
\[ h'(x) = \frac{2}{x} \]

4. **Apply the quotient rule:**
\[ f'(x) = \frac{\left(\frac{1}{x \log x}\right)(2 \log x) - \log(\log x)\left(\frac{2}{x}\right)}{(2 \log x)^2} \]
Simplifying the numerator:
\[ = \frac{\frac{2}{x} - \frac{2 \log(\log x)}{x}}{4 (\log x)^2} \]
\[ = \frac{2(1 - \log(\log x))}{4x (\log x)^2} = \frac{1 - \log(\log x)}{2x (\log x)^2} \]

5. **Evaluate \( f'(x) \) at \( x = e \):**
- First, calculate \( \log e = 1 \) and \( \log(\log e) = \log(1) = 0 \).
\[ f'(e) = \frac{1 - 0}{2e(1)^2} = \frac{1}{2e} \]

### Final Answer:
\[ f'(e) = \frac{1}{2e} \]