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

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

Options:

Correct Answer: 2xe^(x^2)

Solution:

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

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

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

Correct Answer: 2x e^(x^2)

Step 1: Identify the function you want to differentiate, which is f(x) = e^(x^2).
Step 2: Recognize that this is a composite function, where the outer function is e^u (with u = x^2) and the inner function is u = x^2.
Step 3: Apply the chain rule, which states that the derivative of e^u with respect to x is e^u * (du/dx).
Step 4: Find the derivative of the inner function u = x^2. The derivative du/dx = 2x.
Step 5: Substitute back into the chain rule formula: f'(x) = e^(x^2) * (du/dx).
Step 6: Replace du/dx with 2x: f'(x) = e^(x^2) * 2x.
Step 7: Rearrange the expression to get the final answer: f'(x) = 2x e^(x^2).

Chain Rule – The chain rule is a fundamental differentiation technique used to find the derivative of composite functions.
Exponential Functions – Understanding the properties of exponential functions, particularly how they behave under differentiation.