If f(x) = e^(2x), what is f'(x)?

If f(x) = e^(2x), what is f'(x)?

Correct Answer: 2e^(2x)

Step 1: Identify the function f(x) = e^(2x).
Step 2: Recognize that this is an exponential function where the exponent is a function of x (specifically, 2x).
Step 3: Recall the chain rule for differentiation, which states that if you have a function g(h(x)), the derivative is g'(h(x)) * h'(x).
Step 4: In our case, g(u) = e^u where u = 2x. The derivative of g(u) = e^u is g'(u) = e^u.
Step 5: Now find h'(x) where h(x) = 2x. The derivative h'(x) = 2.
Step 6: Apply the chain rule: f'(x) = g'(h(x)) * h'(x) = e^(2x) * 2.
Step 7: Simplify the expression: f'(x) = 2e^(2x).

Differentiation of Exponential Functions – Understanding how to differentiate functions of the form f(x) = e^(kx) using the chain rule.