If ` f (x) = ln _(x) `(ln x ), then f'(e) =

Options

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

Answer

To find \( f'(e) \) for the function \( f(x) = \ln(\ln(x)) \), we will follow these steps:

### Step 1: Differentiate the function
Given:
\[ f(x) = \ln(\ln(x)) \]
To differentiate \( f(x) \), we will use the chain rule. The derivative of \( \ln(u) \) is \( \frac{1}{u} \cdot u' \), where \( u = \ln(x) \).

So, we first find the derivative of \( \ln(x) \):
\[ u = \ln(x) \implies u' = \frac{1}{x} \]

Now, applying the chain rule:
\[ f'(x) = \frac{1}{\ln(x)} \cdot \frac{1}{x} \]
Thus, we have:
\[ f'(x) = \frac{1}{x \ln(x)} \]

### Step 2: Evaluate the derivative at \( x = e \)
Now we need to find \( f'(e) \):
\[ f'(e) = \frac{1}{e \ln(e)} \]
Since \( \ln(e) = 1 \), we can substitute this value:
\[ f'(e) = \frac{1}{e \cdot 1} = \frac{1}{e} \]

### Final Answer
Therefore, the value of \( f'(e) \) is:
\[ f'(e) = \frac{1}{e} \]

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