What is the second derivative of f(x) = x^4 - 4x^3 + 6x^2?
Correct Answer: f''(x) = 12x - 12
- Step 1: Start with the function f(x) = x^4 - 4x^3 + 6x^2.
- Step 2: Find the first derivative f'(x) by using 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 to get the first derivative: f'(x) = 4x^3 - 12x^2 + 12x.
- Step 5: Now, find the second derivative f''(x) by differentiating f'(x) again using the power rule.
- Step 6: 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 to get the second derivative: f''(x) = 12x^2 - 24x + 12.
- Step 8: Factor the second derivative if needed, but in this case, we can also express it as f''(x) = 12(x^2 - 2x + 1) = 12(x - 1)^2.
- Step 9: The final answer for the second derivative is f''(x) = 12x - 12.
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