Find the derivative of `f(x) = x^(3) + 1` at x = - 1

Answer

To find the derivative of the function \( f(x) = x^3 + 1 \) at \( x = -1 \), we will follow these steps:

### Step 1: Find the derivative of the function \( f(x) \).

The function is given by:
\[ f(x) = x^3 + 1 \]

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

Applying the power rule to \( f(x) \):
\[ f'(x) = 3x^2 \]

### Step 2: Evaluate the derivative at \( x = -1 \).

Now, we need to find \( f'(-1) \):
\[ f'(-1) = 3(-1)^2 \]

Calculating this gives:
\[ f'(-1) = 3 \cdot 1 = 3 \]

### Final Answer:

Thus, the derivative of \( f(x) = x^3 + 1 \) at \( x = -1 \) is:
\[ \boxed{3} \]