If `f(x)=log_(2)[log_(3)(log_(5)x)]," then "f'(125)=`

Options

A. `(1)/(125log2log3log5)`
B. `(1)/(375log2log3log5)`
C. `(1)/(500log2log3log5)`
D. `(1)/(250log2log3log5)`

Answer

To find \( f'(125) \) for the function \( f(x) = \log_2(\log_3(\log_5(x))) \), we will follow these steps:

### Step 1: Differentiate the function
We will use the chain rule to differentiate \( f(x) \).

1. Start with the outer function \( \log_2(u) \), where \( u = \log_3(\log_5(x)) \).
\[ f'(x) = \frac{1}{\ln(2)} \cdot u' \]

2. Now differentiate \( u = \log_3(v) \), where \( v = \log_5(x) \).
\[ u' = \frac{1}{\ln(3)} \cdot v' \]

3. Next, differentiate \( v = \log_5(x) \).
\[ v' = \frac{1}{\ln(5)} \cdot \frac{1}{x} \]

Combining these derivatives:
\[ f'(x) = \frac{1}{\ln(2)} \cdot \frac{1}{\ln(3)} \cdot \frac{1}{\ln(5)} \cdot \frac{1}{x} \]

### Step 2: Substitute \( x = 125 \)
Now we will substitute \( x = 125 \) into the derivative we found:
\[ f'(125) = \frac{1}{\ln(2) \cdot \ln(3) \cdot \ln(5)} \cdot \frac{1}{125} \]

### Step 3: Simplify the expression
Thus, we can express \( f'(125) \) as:
\[ f'(125) = \frac{1}{125 \cdot \ln(2) \cdot \ln(3) \cdot \ln(5)} \]

### Final Result
The final result for \( f'(125) \) is:
\[ f'(125) = \frac{1}{125 \cdot \ln(2) \cdot \ln(3) \cdot \ln(5)} \]