SUMMARY
The derivative of the composite function f(g(x)) where f(x) = e^(2x) and g(x) = ln(x) is calculated using the chain rule. The first step involves determining f'(x) = 2e^(2x) and g'(x) = 1/x. At x = e, g(e) = 1, leading to f'(g(e)) = f'(1) = 2e^2. Therefore, the derivative at x = e is 2e^2 * (1/e) = 2e.
PREREQUISITES
- Understanding of the chain rule in calculus
- Knowledge of exponential functions and their derivatives
- Familiarity with logarithmic functions and their derivatives
- Basic skills in evaluating derivatives at specific points
NEXT STEPS
- Review the chain rule in calculus for composite functions
- Practice finding derivatives of exponential functions like e^(kx)
- Study the properties and derivatives of logarithmic functions
- Explore applications of derivatives in real-world scenarios
USEFUL FOR
Students studying calculus, particularly those focusing on derivatives and the chain rule, as well as educators looking for examples of composite function differentiation.