If `f(x)=4x^(3)-1`, then `f^(')(x)=…………………. .`

Answer

To find the derivative of the function \( f(x) = 4x^3 - 1 \), we will apply the power rule of differentiation. The power rule states that if \( f(x) = ax^n \), then the derivative \( f'(x) = nax^{n-1} \).

### Step-by-Step Solution:

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

2. **Differentiate each term**:
- The first term is \( 4x^3 \):
- Using the power rule:
\[ \frac{d}{dx}(4x^3) = 4 \cdot 3x^{3-1} = 12x^2 \]
- The second term is \( -1 \):
- The derivative of a constant is 0:
\[ \frac{d}{dx}(-1) = 0 \]

3. **Combine the results**:
\[ f'(x) = 12x^2 + 0 = 12x^2 \]

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