Differentiate the function f(x) = x^2 * e^x.

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Question: Differentiate the function f(x) = x^2 * e^x.

Options:

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

Solution:

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

Differentiate the function f(x) = x^2 * e^x.

Differentiate the function f(x) = x^2 * e^x.

Step 1: Identify the function to differentiate, which is f(x) = x^2 * e^x.
Step 2: Recognize that this is a product of two functions: u = x^2 and v = e^x.
Step 3: Recall the product rule for differentiation: (u * v)' = u' * v + u * v'.
Step 4: Differentiate u = x^2. The derivative u' = 2x.
Step 5: Differentiate v = e^x. The derivative v' = e^x (since the derivative of e^x is e^x).
Step 6: Apply the product rule: f'(x) = u' * v + u * v'.
Step 7: Substitute the derivatives into the product rule: f'(x) = (2x) * e^x + (x^2) * (e^x).
Step 8: Simplify the expression: f'(x) = 2x * e^x + x^2 * e^x.

Product Rule – The product rule is a formula used to find the derivative of the product of two functions.
Exponential Function Derivative – The derivative of the exponential function e^x is e^x itself.
Polynomial Derivative – The derivative of a polynomial function, such as x^2, is found using the power rule.