derivative of `log(x^2+3)`

Answer

To find the derivative of the function \( f(x) = \log(x^2 + 3) \), we will use the chain rule for differentiation.

### Step-by-Step Solution:

1. **Identify the outer and inner functions**:
The outer function is \( \log(u) \) where \( u = x^2 + 3 \). The inner function is \( u = x^2 + 3 \).

2. **Differentiate the outer function**:
The derivative of \( \log(u) \) with respect to \( u \) is \( \frac{1}{u} \).

3. **Differentiate the inner function**:
The derivative of \( u = x^2 + 3 \) with respect to \( x \) is \( \frac{du}{dx} = 2x \), since the derivative of \( x^2 \) is \( 2x \) and the derivative of a constant (3) is 0.

4. **Apply the chain rule**:
According to the chain rule, the derivative of \( f(x) \) is given by:
\[ f'(x) = \frac{d}{dx} \log(u) = \frac{1}{u} \cdot \frac{du}{dx} \]
Substituting \( u \) and \( \frac{du}{dx} \) into the equation, we have:
\[ f'(x) = \frac{1}{x^2 + 3} \cdot 2x \]

5. **Simplify the expression**:
Therefore, the derivative of \( f(x) = \log(x^2 + 3) \) is:
\[ f'(x) = \frac{2x}{x^2 + 3} \]

### Final Answer:
The derivative of \( \log(x^2 + 3) \) is \( \frac{2x}{x^2 + 3} \).

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