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

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

Options:

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

Solution:

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

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

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

Step 1: Identify the function f(x) = x^2 * e^x. This is a product of two functions: u = x^2 and v = e^x.
Step 2: Recall the product rule for derivatives. The product rule states that if you have two functions u and v, then the derivative f'(x) = u'v + uv'.
Step 3: Find the derivative of u = x^2. The derivative u' = 2x.
Step 4: Find the derivative of v = e^x. The derivative v' = e^x (since the derivative of e^x is e^x).
Step 5: Apply the product rule: f'(x) = u'v + uv' = (2x)(e^x) + (x^2)(e^x).
Step 6: Factor out e^x from both terms: f'(x) = e^x(2x + x^2).
Step 7: Rearrange the expression: f'(x) = e^x(x^2 + 2x).

Product Rule – The product rule is a formula used to find the derivative of the product of two functions.
Exponential Functions – Understanding how to differentiate exponential functions, such as e^x.
Polynomial Functions – Differentiating polynomial functions, such as x^2.