If h(x) = e^(2x), what is h'(x)? (2019)

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Question: If h(x) = e^(2x), what is h\'(x)? (2019)

Options:

Correct Answer: 2e^(2x)

Exam Year: 2019

Solution:

Using the chain rule, h\'(x) = 2e^(2x).

If h(x) = e^(2x), what is h'(x)? (2019)

If h(x) = e^(2x), what is h'(x)? (2019)

Step 1: Identify the function h(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 composite function f(g(x)), the derivative is f'(g(x)) * g'(x).
Step 4: In our case, let f(u) = e^u where u = 2x. The derivative of f(u) with respect to u is f'(u) = e^u.
Step 5: Now, find the derivative of u = 2x with respect to x, which is u' = 2.
Step 6: Apply the chain rule: h'(x) = f'(u) * u' = e^(2x) * 2.
Step 7: Simplify the expression: h'(x) = 2e^(2x).

Differentiation of Exponential Functions – Understanding how to differentiate functions of the form h(x) = e^(kx) using the chain rule.
Chain Rule – Applying the chain rule correctly when differentiating composite functions.