If f(x) ` = log_(x^2) (log x)` then `f' (x) ` at x=e is

Options

A. 0
B. 1
C. `1//E`
D. `1//2e`

Answer

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

### Step 1: Rewrite the logarithm
Using the change of base formula for logarithms, we can rewrite \( f(x) \):
\[ f(x) = \frac{\log(\log x)}{\log(x^2)} \]
Since \( \log(x^2) = 2\log x \), we can simplify this to:
\[ f(x) = \frac{\log(\log x)}{2 \log x} \]

### Step 2: Differentiate using the quotient rule
To find \( f'(x) \), we will use the quotient rule. If \( u = \log(\log x) \) and \( v = 2 \log x \), then:
\[ f'(x) = \frac{u'v - uv'}{v^2} \]
We need to find \( u' \) and \( v' \).

### Step 3: Find \( u' \) and \( v' \)
1. For \( u = \log(\log x) \):
\[ u' = \frac{1}{\log x} \cdot \frac{1}{x} = \frac{1}{x \log x} \]
2. For \( v = 2 \log x \):
\[ v' = 2 \cdot \frac{1}{x} = \frac{2}{x} \]

### Step 4: Substitute \( u, u', v, v' \) into the quotient rule
Now substituting these into the quotient rule:
\[ f'(x) = \frac{\left(\frac{1}{x \log x}\right)(2 \log x) - (\log(\log x))\left(\frac{2}{x}\right)}{(2 \log x)^2} \]

### Step 5: Simplify the expression
This simplifies to:
\[ f'(x) = \frac{\frac{2}{x} - \frac{2 \log(\log x)}{x}}{4 (\log x)^2} \]
Factoring out \( \frac{2}{x} \):
\[ f'(x) = \frac{2}{x} \cdot \frac{1 - \log(\log x)}{4 (\log x)^2} = \frac{1 - \log(\log x)}{2x (\log x)^2} \]

### Step 6: Evaluate at \( x = e \)
Now we will evaluate \( f'(x) \) at \( x = e \):
1. Calculate \( \log e = 1 \).
2. Calculate \( \log(\log e) = \log(1) = 0 \).

Substituting these values:
\[ f'(e) = \frac{1 - 0}{2e(1)^2} = \frac{1}{2e} \]

### Final Result
Thus, the value of \( f'(e) \) is:
\[ \boxed{\frac{1}{2e}} \]