If `f(x)=log_(e)[log_(e)x]`, then what is f' (e) equal to?

Options

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

Answer

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

### Step-by-Step Solution:

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

2. **Differentiate using the chain rule**:
We will apply the chain rule to differentiate \( f(x) \). The derivative of \( \log_e u \) is \( \frac{1}{u} \cdot \frac{du}{dx} \).

Let \( u = \log_e x \). Then,
\[ f(x) = \log_e(u) \]

The derivative \( f'(x) \) is:
\[ f'(x) = \frac{1}{u} \cdot \frac{du}{dx} \]

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

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

5. **Evaluate \( f'(e) \)**:
Now we need to evaluate \( f'(x) \) at \( x = e \):
\[ f'(e) = \frac{1}{e \log_e e} \]

Since \( \log_e e = 1 \):
\[ f'(e) = \frac{1}{e \cdot 1} = \frac{1}{e} \]

### Final Answer:
Thus, \( f'(e) = \frac{1}{e} \).

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