Calculate the derivative of f(x) = e^(2x).

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Question: Calculate the derivative of f(x) = e^(2x).

Options:

Correct Answer: 2e^(2x)

Solution:

Using the chain rule, f\'(x) = d/dx(e^(2x)) = 2e^(2x).

Calculate the derivative of f(x) = e^(2x).

Calculate the derivative of f(x) = e^(2x).

Correct Answer: 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: Differentiate the outer function e^u. The derivative of e^u is e^u itself.
Step 5: Differentiate the inner function u = 2x. The derivative of 2x is 2.
Step 6: Combine the results from the chain rule: f'(x) = e^(2x) * 2.
Step 7: Simplify the expression: f'(x) = 2e^(2x).

Chain Rule – The chain rule is a formula for computing the derivative of the composition of two or more functions.
Exponential Functions – Understanding the properties of exponential functions, particularly how their derivatives behave.