SUMMARY
The derivative of e^{2x} can be obtained from first principles using the limit definition of the derivative. The expression simplifies to e^{2x}(\frac{e^{2h}-1}{h}) as h approaches 0. By substituting u = 2h, the limit can be expressed as 2\frac{e^u - 1}{u}, which approaches 2 as u approaches 0. Thus, the derivative of e^{2x} is confirmed to be 2e^{2x}.
PREREQUISITES
- Understanding of limits in calculus
- Familiarity with the exponential function e^x
- Knowledge of the derivative definition from first principles
- Basic algebraic manipulation skills
NEXT STEPS
- Study the limit definition of derivatives in calculus
- Explore the properties of the exponential function e^x
- Learn about L'Hôpital's Rule for evaluating limits
- Practice finding derivatives of other exponential functions
USEFUL FOR
Students studying calculus, particularly those focusing on derivatives and limits, as well as educators looking for examples of first principles differentiation.