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Derivative of x³ + 2x² - 5x + 3
Published on October 7, 2024
Calculus
Find the derivative of the polynomial function f(x) = x³ + 2x² - 5x + 3 using the power rule. The derivative is f'(x) = 3x² + 4x - 5.
Question
Find the derivative of f(x) = x³ + 2x² - 5x + 3
Answer
Here’s how to find the derivative of f(x) = x³ + 2x² - 5x + 3 step-by-step:
1. Recall the Power Rule:
The power rule states that the derivative of x<sup>n</sup> is nx<sup>n-1</sup>. We’ll use this rule for each term in the function.
2. Apply the Power Rule to Each Term:
- Term 1: The derivative of x³ is 3x².
- Term 2: The derivative of 2x² is (2)(2)x¹ = 4x.
- Term 3: The derivative of -5x is -5.
- Term 4: The derivative of 3 (a constant) is 0.
3. Combine the Derivatives:
The derivative of the entire function is the sum of the derivatives of each term:
f’(x) = 3x² + 4x - 5
Therefore, the derivative of f(x) = x³ + 2x² - 5x + 3 is f’(x) = 3x² + 4x - 5.