if `f(x) = log_5 log_3 x,` then `f '(e ) ` is

Options

A. `e log 5`
B. ` -e log 5`
C. ` (1)/( e log 5)`
D. None

Answer

To find \( f'(e) \) for the function \( f(x) = \log_5(\log_3(x)) \), we will follow these steps:

### Step 1: Rewrite the logarithmic function using change of base formula
Using the change of base formula, we can express \( f(x) \) as:
\[ f(x) = \frac{\log_e(\log_3(x))}{\log_e(5)} \]
This simplifies our differentiation process since we are using the natural logarithm \( \log_e \).

### Step 2: Differentiate \( f(x) \)
Now we differentiate \( f(x) \) with respect to \( x \):
\[ f'(x) = \frac{1}{\log_e(5)} \cdot \frac{d}{dx}[\log_e(\log_3(x))] \]
To differentiate \( \log_e(\log_3(x)) \), we apply the chain rule:
\[ \frac{d}{dx}[\log_e(\log_3(x))] = \frac{1}{\log_3(x)} \cdot \frac{d}{dx}[\log_3(x)] \]
Next, we differentiate \( \log_3(x) \):
\[ \frac{d}{dx}[\log_3(x)] = \frac{1}{x \cdot \log_e(3)} \]
Combining these, we have:
\[ \frac{d}{dx}[\log_e(\log_3(x))] = \frac{1}{\log_3(x)} \cdot \frac{1}{x \cdot \log_e(3)} = \frac{1}{x \cdot \log_3(x) \cdot \log_e(3)} \]
Thus, the derivative \( f'(x) \) becomes:
\[ f'(x) = \frac{1}{\log_e(5)} \cdot \frac{1}{x \cdot \log_3(x) \cdot \log_e(3)} \]

### Step 3: Evaluate \( f'(e) \)
Now we substitute \( x = e \) into \( f'(x) \):
\[ f'(e) = \frac{1}{\log_e(5)} \cdot \frac{1}{e \cdot \log_3(e) \cdot \log_e(3)} \]
Next, we need to evaluate \( \log_3(e) \):
Using the change of base formula again:
\[ \log_3(e) = \frac{\log_e(e)}{\log_e(3)} = \frac{1}{\log_e(3)} \]
Substituting this back into our expression for \( f'(e) \):
\[ f'(e) = \frac{1}{\log_e(5)} \cdot \frac{1}{e \cdot \frac{1}{\log_e(3)} \cdot \log_e(3)} = \frac{1}{\log_e(5)} \cdot \frac{1}{e} \]
Thus, we simplify to:
\[ f'(e) = \frac{1}{e \cdot \log_e(5)} \]

### Final Answer
\[ f'(e) = \frac{1}{e \cdot \log_e(5)} \]