If f(x)=logx , then the derivative of f (logx)w.r.t.x is

Options

A. `(logx)/(x)`
B. `(x)/(logx)`
C. `xlogx`
D. `(1)/(xlogx)`

Answer

To find the derivative of \( f(\log x) \) with respect to \( x \), where \( f(x) = \log x \), we can follow these steps:

### Step 1: Substitute \( f(x) \)
Given \( f(x) = \log x \), we need to find \( f(\log x) \):
\[ f(\log x) = \log(\log x) \]

### Step 2: Differentiate \( f(\log x) \)
Now, we need to differentiate \( f(\log x) = \log(\log x) \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx} \log(\log x) \]

### Step 3: Apply the Chain Rule
Using the chain rule, we differentiate \( \log(\log x) \):
\[ \frac{dy}{dx} = \frac{1}{\log x} \cdot \frac{d}{dx}(\log x) \]

### Step 4: Differentiate \( \log x \)
Now, we find the derivative of \( \log x \):
\[ \frac{d}{dx}(\log x) = \frac{1}{x} \]

### Step 5: Combine the Results
Substituting back into the derivative:
\[ \frac{dy}{dx} = \frac{1}{\log x} \cdot \frac{1}{x} \]
Thus, we have:
\[ \frac{dy}{dx} = \frac{1}{x \log x} \]

### Final Result
The derivative of \( f(\log x) \) with respect to \( x \) is:
\[ \frac{dy}{dx} = \frac{1}{x \log x} \]