Overview
This lecture demonstrates how to find the derivative of ( f(x) = 2x^2 e^{5x} ) using the product rule, including step-by-step differentiation and factorization.
Identifying Components for the Product Rule
- Recognize the function as a product: ( f(x) = 2x^2 ) and ( g(x) = e^{5x} ).
- Set up the product rule: ( [f(x)g(x)]' = f(x)g'(x) + g(x)f'(x) ).
Applying the Product Rule
- Find ( g'(x) ): The derivative of ( e^{5x} ) is ( e^{5x} \cdot 5 ) by the chain rule.
- Find ( f'(x) ): The derivative of ( 2x^2 ) is ( 4x ).
- Substitute into the product rule:
( 2x^2 \cdot 5e^{5x} + e^{5x} \cdot 4x ).
Simplifying and Factoring the Final Answer
- Multiply out terms: ( 10x^2 e^{5x} + 4x e^{5x} ).
- Factor the result: Factor out ( 2x e^{5x} ) to get ( 2x e^{5x}(5x + 2) ).
Key Terms & Definitions
- Product Rule — If ( h(x) = f(x)g(x) ), then ( h'(x) = f(x)g'(x) + g(x)f'(x) ).
- Chain Rule — The derivative of ( e^{u(x)} ) is ( e^{u(x)} \cdot u'(x) ).
- Factoring — Expressing an expression as a product of its greatest common factors.
Action Items / Next Steps
- Review the product rule and chain rule for exponential functions.
- Prepare for the next example involving the exponential function and the quotient rule.