If `f(x) = log _ex` then the differential coefficent of `f(log_ex)` with respect to x is

Options

A. `x/log_ex`
B. `log_ex/x`
C. `1/(xlog_ex)`
D. none of these

Answer

To solve the problem, we need to find the differential coefficient of \( f(\log_e x) \) where \( f(x) = \log_e x \).

### Step-by-Step Solution:

1. **Identify the function**: We have \( f(x) = \log_e x \). We need to evaluate \( f(\log_e x) \).

\[ f(\log_e x) = \log_e(\log_e x) \]

2. **Differentiate \( f(\log_e x) \)**: We will use the chain rule for differentiation. The chain rule states that if you have a composite function \( g(h(x)) \), then the derivative is given by \( g'(h(x)) \cdot h'(x) \).

Let \( u = \log_e x \). Then, we need to differentiate \( \log_e u \) with respect to \( x \).

\[ \frac{d}{dx}[\log_e(\log_e x)] = \frac{1}{\log_e x} \cdot \frac{d}{dx}[\log_e x] \]

3. **Differentiate \( \log_e x \)**: The derivative of \( \log_e x \) with respect to \( x \) is:

\[ \frac{d}{dx}[\log_e x] = \frac{1}{x} \]

4. **Combine the derivatives**: Now we substitute back into our differentiation:

\[ \frac{d}{dx}[\log_e(\log_e x)] = \frac{1}{\log_e x} \cdot \frac{1}{x} \]

Therefore, we have:

\[ \frac{d}{dx}[\log_e(\log_e x)] = \frac{1}{x \log_e x} \]

5. **Final Result**: The differential coefficient of \( f(\log_e x) \) with respect to \( x \) is:

\[ \frac{1}{x \log_e x} \]

### Conclusion:
The answer is \( \frac{1}{x \log_e x} \), which corresponds to option C.