Calculate the derivative of f(x) = x^2 * e^x. (2023) 2023

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Question: Calculate the derivative of f(x) = x^2 * e^x. (2023) 2023

Options:

Correct Answer: e^x(2x + 1)

Exam Year: 2023

Solution:

Using the product rule, f\'(x) = d/dx(x^2 * e^x) = e^x(2x + 1).

Calculate the derivative of f(x) = x^2 * e^x. (2023) 2023

Calculate the derivative of f(x) = x^2 * e^x. (2023) 2023

Step 1: Identify the function 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 derivatives: if f(x) = u * v, then f'(x) = u' * v + u * v'.
Step 4: Calculate the derivative of u: u' = d/dx(x^2) = 2x.
Step 5: Calculate the derivative of v: v' = d/dx(e^x) = e^x.
Step 6: Apply the product rule: f'(x) = u' * v + u * v' = (2x) * (e^x) + (x^2) * (e^x).
Step 7: Factor out e^x from both terms: f'(x) = e^x(2x + x^2).
Step 8: Simplify the expression: f'(x) = e^x(2x + 1).

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.