SUMMARY
The derivative of the function f(x) = 2xe^(-x) is f'(x) = (2 - 2x)e^(-x). The discussion emphasizes the application of the product rule for differentiation and simplifies the derivative effectively. The critical point identified is x = 1, which is confirmed to be a maximum value of the function. This conclusion is reached by setting the derivative to zero and analyzing the behavior of the function at this point.
PREREQUISITES
- Understanding of calculus, specifically differentiation techniques.
- Familiarity with the product rule and chain rule in calculus.
- Knowledge of exponential functions and their derivatives.
- Ability to analyze critical points and determine maxima and minima.
NEXT STEPS
- Study the product rule and chain rule in more depth.
- Learn about critical points and how to classify them using the first and second derivative tests.
- Explore the behavior of exponential functions in calculus.
- Practice finding derivatives of more complex functions involving products and exponentials.
USEFUL FOR
Students and professionals in mathematics, particularly those studying calculus, as well as educators looking to reinforce concepts of differentiation and critical point analysis.