The derivative of f(x) = x log x is

Options

A. `(x)/(log x)`
B. 1
C. 1 + log x
D. 1 - log x

Answer

To find the derivative of the function \( f(x) = x \log x \), we will use the product rule of differentiation. The product rule states that if you have two functions \( u \) and \( v \), then the derivative of their product \( uv \) is given by:

\[ \frac{d}{dx}(uv) = u'v + uv' \]

where \( u' \) is the derivative of \( u \) and \( v' \) is the derivative of \( v \).

### Step-by-Step Solution:

1. **Identify the functions**:
Let \( u = x \) and \( v = \log x \).

2. **Differentiate \( u \) and \( v \)**:
- The derivative of \( u \) with respect to \( x \) is:
\[ u' = \frac{d}{dx}(x) = 1 \]
- The derivative of \( v \) with respect to \( x \) is:
\[ v' = \frac{d}{dx}(\log x) = \frac{1}{x} \]

3. **Apply the product rule**:
Using the product rule:
\[ f'(x) = u'v + uv' \]
Substitute \( u \), \( v \), \( u' \), and \( v' \):
\[ f'(x) = (1)(\log x) + (x)\left(\frac{1}{x}\right) \]

4. **Simplify the expression**:
The second term simplifies as follows:
\[ f'(x) = \log x + 1 \]

5. **Final result**:
Therefore, the derivative of \( f(x) = x \log x \) is:
\[ f'(x) = 1 + \log x \]