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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
To find the derivative of f(x) = x³ + 2x² - 5x + 3, we’ll use the power rule. The power rule states:
d/dx (x^n) = n*x^(n-1)
Let’s break it down step-by-step:
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Derivative of x³:
- Applying the power rule: d/dx (x³) = 3x^(3-1) = 3x²
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Derivative of 2x²:
- Applying the power rule: d/dx (2x²) = 2 * 2x^(2-1) = 4x
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Derivative of -5x:
- Applying the power rule: d/dx (-5x) = -5 * 1x^(1-1) = -5
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Derivative of 3:
- The derivative of a constant is always zero: d/dx (3) = 0
Now, we add up the derivatives of each term to find the derivative of the entire function:
f’(x) = 3x² + 4x - 5 + 0
Therefore, the derivative of f(x) = x³ + 2x² - 5x + 3 is:
f’(x) = 3x² + 4x - 5