The differential coefficient of f ( log x) where f (x) =log x ` is

Options

A. `x//log x `
B. ` ( log x ) //x `
C. `(x log x )^(-1)`
D. None of these

Answer

To find the differential coefficient of \( f(\log x) \) where \( f(x) = \log x \), we will follow these steps:

### Step 1: Identify the function
We have \( f(x) = \log x \). We need to find \( f(\log x) \).

### Step 2: Substitute \( \log x \) into \( f \)
Substituting \( \log x \) into the function gives us:
\[ f(\log x) = \log(\log x) \]

### Step 3: Differentiate \( f(\log x) \)
Now we need to differentiate \( f(\log x) = \log(\log x) \) with respect to \( x \). We will use the chain rule for differentiation.

### Step 4: Apply the chain rule
Using the chain rule:
\[ \frac{d}{dx} \log(\log x) = \frac{1}{\log x} \cdot \frac{d}{dx}(\log x) \]

### Step 5: Differentiate \( \log x \)
Now, we differentiate \( \log x \):
\[ \frac{d}{dx}(\log x) = \frac{1}{x} \]

### Step 6: Combine the results
Substituting back into our differentiation gives:
\[ \frac{d}{dx} \log(\log x) = \frac{1}{\log x} \cdot \frac{1}{x} \]

### Step 7: Final expression
Thus, the differential coefficient of \( f(\log x) \) is:
\[ \frac{1}{x \log x} \]

### Summary
The differential coefficient of \( f(\log x) \) where \( f(x) = \log x \) is:
\[ \frac{1}{x \log x} \]

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