If `f(x)=log_(a)(log_(a)x)`, then f'(x), is

Options

A. `(log_(a)e)/(xlog_(e)x)`
B. `(log_(e)a)/(xlog_(a)x)`
C. `(log_(e)a)/(x)`
D. `(x)/(log_(e)a)`

Answer

To find the derivative of the function \( f(x) = \log_a(\log_a(x)) \), we will use the chain rule and the properties of logarithms. Here’s a step-by-step solution:

### Step 1: Rewrite the function using natural logarithms
We know that:
\[ \log_a(x) = \frac{\ln(x)}{\ln(a)} \]
Thus, we can rewrite \( f(x) \):
\[ f(x) = \log_a(\log_a(x)) = \log_a\left(\frac{\ln(x)}{\ln(a)}\right) \]

### Step 2: Apply the logarithm property
Using the property of logarithms, we can express \( f(x) \) as:
\[ f(x) = \frac{\ln\left(\frac{\ln(x)}{\ln(a)}\right)}{\ln(a)} \]
This simplifies to:
\[ f(x) = \frac{\ln(\ln(x)) - \ln(\ln(a))}{\ln(a)} \]

### Step 3: Differentiate using the chain rule
Now we differentiate \( f(x) \):
\[ f'(x) = \frac{1}{\ln(a)} \cdot \frac{d}{dx}[\ln(\ln(x))] \]
Using the chain rule, we have:
\[ \frac{d}{dx}[\ln(\ln(x))] = \frac{1}{\ln(x)} \cdot \frac{d}{dx}[\ln(x)] = \frac{1}{\ln(x)} \cdot \frac{1}{x} \]
Thus:
\[ f'(x) = \frac{1}{\ln(a)} \cdot \frac{1}{\ln(x)} \cdot \frac{1}{x} \]

### Step 4: Final expression for the derivative
Combining everything, we get:
\[ f'(x) = \frac{1}{x \ln(x) \ln(a)} \]

### Final Answer
\[ f'(x) = \frac{1}{x \ln(x) \ln(a)} \]

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