To find the second derivative $f''(x)$ of the function $f(x) = (x-1)^3$, follow these steps:

1. First Derivative ($f'(x)$): Use the power rule: $f'(x) = 3(x-1)^2 \cdot (1) = 3(x-1)^2$

2. Second Derivative ($f''(x)$): Differentiate $f'(x) = 3(x-1)^2$ again: $f''(x) = 3 \cdot 2(x-1) \cdot (1) = 6(x-1)$

Final Answer:

$f''(x) = 6(x-1)$

Would you like a detailed explanation of the steps, or do you have specific questions?

Related Questions:

1. What is the third derivative $f'''(x)$ for the same function?
2. How do we verify the correctness of the second derivative?
3. Can you graph $f(x)$, $f'(x)$, and $f''(x)$ to visualize their relationships?
4. How does $f(x)$ change concavity based on $f''(x)$?
5. What are some real-world applications of second derivatives like $f''(x)$?

Tip:

When differentiating composite functions, always remember the chain rule!