What is the derivative of f(x) = x^3 - 3x^2 + 4?
Correct Answer: f'(x) = 3x^2 - 6x
- Step 1: Identify the function f(x) = x^3 - 3x^2 + 4.
- Step 2: Recall the power rule for derivatives: If f(x) = x^n, then f'(x) = n*x^(n-1).
- Step 3: Apply the power rule to the first term x^3: The derivative is 3*x^(3-1) = 3x^2.
- Step 4: Apply the power rule to the second term -3x^2: The derivative is -3*2*x^(2-1) = -6x.
- Step 5: The third term is a constant (4), and the derivative of a constant is 0.
- Step 6: Combine the derivatives from Steps 3, 4, and 5: f'(x) = 3x^2 - 6x + 0.
- Step 7: Simplify the expression: f'(x) = 3x^2 - 6x.
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