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

Options

A. `x // log x `
B. ` log x //x`
C. `1// x.log x`
D. None of these

Answer

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

### Step 1: Identify the function
Given \( f(x) = \log(x) \), we need to find \( f(\log(x)) \).

### Step 2: Substitute into the function
Substituting \( \log(x) \) into \( f \), we get:
\[ f(\log(x)) = \log(\log(x)) \]

### Step 3: Differentiate using the chain rule
To find the derivative of \( \log(\log(x)) \), we will use the chain rule. The derivative of \( \log(u) \) with respect to \( u \) is \( \frac{1}{u} \), and we will also need to differentiate \( \log(x) \) with respect to \( x \).

Let \( u = \log(x) \). Then:
\[ \frac{d}{dx} \log(\log(x)) = \frac{1}{\log(x)} \cdot \frac{d}{dx}(\log(x)) \]

### Step 4: Differentiate \( \log(x) \)
The derivative of \( \log(x) \) is:
\[ \frac{d}{dx}(\log(x)) = \frac{1}{x} \]

### Step 5: Combine the derivatives
Now substituting back, we have:
\[ \frac{d}{dx} \log(\log(x)) = \frac{1}{\log(x)} \cdot \frac{1}{x} \]
Thus, the final result is:
\[ \frac{d}{dx} \log(\log(x)) = \frac{1}{x \log(x)} \]

### Final Answer
The differential coefficient of \( f(\log(x)) \) is:
\[ \frac{1}{x \log(x)} \]

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