If f(x) = x^2 * e^x, find f'(x). (2019)

₹0.0

Question: If f(x) = x^2 * e^x, find f\'(x). (2019)

Options:

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

Exam Year: 2019

Solution:

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

If f(x) = x^2 * e^x, find f'(x). (2019)

If f(x) = x^2 * e^x, find f'(x). (2019)

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: Factor out e^x from both terms: f'(x) = e^x(2x + x^2).
Step 7: Rearrange the terms inside the parentheses: 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 functions involving exponential terms, such as e^x.
Polynomial Functions – Recognizing how to differentiate polynomial terms, such as x^2.