If `f(x)=4x^(3)+7`, then `f^(')(x)=…………………. .`

Answer

To find the derivative of the function \( f(x) = 4x^3 + 7 \), we will use the rules of differentiation.

### Step-by-Step Solution:

1. **Identify the function**:
\[ f(x) = 4x^3 + 7 \]

2. **Differentiate each term separately**:
- The first term is \( 4x^3 \).
- The second term is \( 7 \) (a constant).

3. **Apply the power rule**:
- The power rule states that if \( f(x) = ax^n \), then \( f'(x) = n \cdot ax^{n-1} \).
- For the term \( 4x^3 \):
\[ \text{Differentiate: } f'(x) = 3 \cdot 4x^{3-1} = 12x^2 \]

4. **Differentiate the constant**:
- The derivative of a constant is \( 0 \):
\[ \text{Differentiate: } \frac{d}{dx}(7) = 0 \]

5. **Combine the results**:
- Now, we combine the derivatives of both terms:
\[ f'(x) = 12x^2 + 0 = 12x^2 \]

### Final Answer:
\[ f'(x) = 12x^2 \]