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 follow these steps:

### Step 1: Rewrite the function
Let \( y = f(x) = \log_a(\log_a x) \).

### Step 2: Change of base formula
Using the change of base formula for logarithms, we can rewrite the function:
\[ y = \frac{\log_e(\log_a x)}{\log_e a} \]
where \( \log_e \) is the natural logarithm.

### Step 3: Differentiate using the chain rule
Now we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{1}{\log_e a} \cdot \frac{d}{dx} \left( \log_e(\log_a x) \right) \]

### Step 4: Differentiate the inner function
Next, we need to differentiate \( \log_e(\log_a x) \). Again, we apply the chain rule:
\[ \frac{d}{dx} \left( \log_e(\log_a x) \right) = \frac{1}{\log_a x} \cdot \frac{d}{dx}(\log_a x) \]

### Step 5: Differentiate \( \log_a x \)
Using the change of base formula again, we differentiate \( \log_a x \):
\[ \frac{d}{dx}(\log_a x) = \frac{1}{\log_e a} \cdot \frac{1}{x} \]

### Step 6: Combine the derivatives
Substituting this back into our previous expression:
\[ \frac{d}{dx} \left( \log_e(\log_a x) \right) = \frac{1}{\log_a x} \cdot \left( \frac{1}{\log_e a} \cdot \frac{1}{x} \right) \]
Thus,
\[ \frac{dy}{dx} = \frac{1}{\log_e a} \cdot \frac{1}{\log_a x} \cdot \frac{1}{\log_e a} \cdot \frac{1}{x} \]
This simplifies to:
\[ \frac{dy}{dx} = \frac{1}{(\log_e a)^2} \cdot \frac{1}{x \log_a x} \]

### Step 7: Final expression
Therefore, the derivative \( f'(x) \) is:
\[ f'(x) = \frac{1}{(\log_e a)^2} \cdot \frac{1}{x \log_a x} \]