Find the derivative of f(x) = e^(2x).
Correct Answer: f'(x) = 2e^(2x)
- Step 1: Identify the function you want to differentiate, which is f(x) = e^(2x).
- Step 2: Recognize that this function is a composition of two functions: the outer function e^u (where u = 2x) and the inner function u = 2x.
- Step 3: Apply the chain rule, which states that the derivative of e^u with respect to x is e^u times the derivative of u with respect to x.
- Step 4: Find the derivative of the outer function e^u, which is e^(2x).
- Step 5: Find the derivative of the inner function u = 2x, which is 2.
- Step 6: Multiply the derivative of the outer function by the derivative of the inner function: f'(x) = e^(2x) * 2.
- Step 7: Simplify the expression to get f'(x) = 2e^(2x).
- Exponential Functions – Understanding the properties of exponential functions and their derivatives.
- Chain Rule – Applying the chain rule correctly when differentiating composite functions.
