Differentiate the following w.r.t.x. The differentiation coneffiecient of `f(log_(e)x)w.r.t.x,` where `f(x)=log_(e)x,` is

Options

A. `(x)/(log_(e)x)`
B. `(1)/(x)log_(e)x`
C. `(1)/(xlog_(e)x)`
D. None of these

Answer

To differentiate the function \( f(\log_e x) \) where \( f(x) = \log_e x \), we will follow these steps:

### Step 1: Define the function
We start with the function:
\[ f(x) = \log_e x \]
We need to find \( f(\log_e x) \):
\[ f(\log_e x) = \log_e(\log_e x) \]

### Step 2: Differentiate using the chain rule
To differentiate \( f(\log_e x) \), we apply the chain rule. The chain rule states that if you have a composite function \( f(g(x)) \), then the derivative is given by:
\[ \frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x) \]
In our case, \( g(x) = \log_e x \).

### Step 3: Find the derivatives
1. **Differentiate \( f(x) = \log_e x \)**:
\[ f'(x) = \frac{1}{x} \]

2. **Differentiate \( g(x) = \log_e x \)**:
\[ g'(x) = \frac{1}{x} \]

### Step 4: Apply the chain rule
Now we apply the chain rule:
\[ \frac{d}{dx} f(\log_e x) = f'(\log_e x) \cdot g'(x) \]
Substituting the derivatives we found:
\[ = f'(\log_e x) \cdot \frac{1}{x} \]
Since \( f'(\log_e x) = \frac{1}{\log_e x} \):
\[ = \frac{1}{\log_e x} \cdot \frac{1}{x} \]

### Step 5: Simplify the expression
Thus, we have:
\[ \frac{d}{dx} f(\log_e x) = \frac{1}{x \cdot \log_e x} \]

### Final Answer
The differentiation coefficient of \( f(\log_e x) \) with respect to \( x \) is:
\[ \frac{1}{x \cdot \log_e x} \]