Find the second derivative of f(x) = x^4 - 4x^3 + 6x^2.

Find the second derivative of f(x) = x^4 - 4x^3 + 6x^2.

Correct Answer: 12x^2 - 24

Step 1: Start with the function f(x) = x^4 - 4x^3 + 6x^2.
Step 2: To find the first derivative, use the power rule. The power rule states that if f(x) = x^n, then f'(x) = n*x^(n-1).
Step 3: Apply the power rule to each term in f(x):
- For x^4, the derivative is 4*x^(4-1) = 4x^3.
- For -4x^3, the derivative is -4*3*x^(3-1) = -12x^2.
- For 6x^2, the derivative is 6*2*x^(2-1) = 12x.
Step 4: Combine the derivatives from each term to get the first derivative: f'(x) = 4x^3 - 12x^2 + 12x.
Step 5: Now, find the second derivative by differentiating f'(x).
Step 6: Again, apply the power rule to each term in f'(x):
- For 4x^3, the derivative is 4*3*x^(3-1) = 12x^2.
- For -12x^2, the derivative is -12*2*x^(2-1) = -24x.
- For 12x, the derivative is 12.
Step 7: Combine the derivatives from each term to get the second derivative: f''(x) = 12x^2 - 24x + 12.
Step 8: The final answer for the second derivative is f''(x) = 12x^2 - 24x + 12.

Differentiation – The process of finding the derivative of a function, which measures how the function changes as its input changes.
Higher-order derivatives – The derivatives of a function beyond the first derivative, specifically the second derivative in this case, which provides information about the curvature of the function.