Overview
This lecture explains how to find the derivative of a quotient involving an exponential function with base e and a polynomial using the quotient rule and chain rule.
Applying the Quotient Rule
- The quotient rule for derivatives is used when differentiating a function that's a quotient of two functions, f(x)/g(x).
- The rule is: [f(x)/g(x)]' = [g(x)f'(x) - f(x)g'(x)] / [g(x)]².
- Here, let the numerator be f(x) and the denominator be g(x).
Differentiating the Numerator and Denominator
- The denominator squared is (x³ + 2)².
- For the numerator: first term is (x³ + 2) × [derivative of e^(2x² - 1)].
- The derivative of e^(2x² - 1) uses the chain rule: e^(2x² - 1) × 4x.
- Second term: subtract [e^(2x² - 1)] × [derivative of x³ + 2], which is 3x².
Simplifying and Factoring the Result
- After expanding, the numerator is 4x⁴e^(2x² - 1) + 8xe^(2x² - 1) - 3x²e^(2x² - 1).
- Factor out the greatest common factor: x e^(2x² - 1).
- The factored numerator is x e^(2x² - 1) (4x³ - 3x + 8).
- The derivative has denominator (x³ + 2)².
Key Terms & Definitions
- Quotient Rule — A rule to differentiate the quotient of two functions: ([g(x)f'(x) - f(x)g'(x)] / [g(x)]²).
- Chain Rule — A rule to differentiate composite functions: derivative of outside × derivative of inside.
Action Items / Next Steps
- Review the steps of the quotient and chain rules.
- Prepare for the next example involving the exponential function and the quotient rule.