If `f(x)=log(logx)," then "f'(e)=`

Options

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

Answer

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

### Step-by-Step Solution:

1. **Identify the function**:
\[ f(x) = \log(\log x) \]

2. **Differentiate using the chain rule**:
To differentiate \( f(x) \), we apply the chain rule. The outer function is \( \log(u) \) where \( u = \log x \). The derivative of \( \log(u) \) is:
\[ f'(x) = \frac{1}{u} \cdot \frac{du}{dx} \]
where \( u = \log x \).

3. **Find \( \frac{du}{dx} \)**:
The derivative of \( u = \log x \) is:
\[ \frac{du}{dx} = \frac{1}{x} \]

4. **Substituting back into the derivative**:
Now substituting \( u \) back into the derivative:
\[ f'(x) = \frac{1}{\log x} \cdot \frac{1}{x} = \frac{1}{x \log x} \]

5. **Evaluate \( f'(e) \)**:
Now we need to evaluate \( f'(e) \):
\[ f'(e) = \frac{1}{e \log e} \]
Since \( \log e = 1 \):
\[ f'(e) = \frac{1}{e \cdot 1} = \frac{1}{e} \]

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