Differentiate f(x) = x^2 * e^x. (2022)

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Question: Differentiate f(x) = x^2 * e^x. (2022)

Options:

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

Exam Year: 2022

Solution:

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

Differentiate f(x) = x^2 * e^x. (2022)

Differentiate f(x) = x^2 * e^x. (2022)

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 differentiation. The product rule states that if you have two functions u and v, then the derivative f'(x) = u'v + uv'.
Step 3: Differentiate u = x^2. The derivative u' = 2x.
Step 4: Differentiate 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: Simplify the expression: f'(x) = 2x * e^x + x^2 * e^x.
Step 7: Factor out e^x: f'(x) = e^x(2x + x^2).

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 e^x is e^x, which is important when differentiating functions involving e.
Polynomial Derivative – The derivative of x^n is n*x^(n-1), which applies to the x^2 term in the function.