Question: Find the derivative of f(x) = x^2 * e^x.
Options:
e^x(x^2 + 2x)
e^x(x^2 - 2x)
2xe^x
x^2e^x
Correct Answer: e^x(x^2 + 2x)
Solution:
Using the product rule: f\'(x) = x^2 * e^x + 2x * e^x = e^x(x^2 + 2x).
Find the derivative of f(x) = x^2 * e^x.
Practice Questions
Q1
Find the derivative of f(x) = x^2 * e^x.
e^x(x^2 + 2x)
e^x(x^2 - 2x)
2xe^x
x^2e^x
Questions & Step-by-Step Solutions
Find the derivative of f(x) = x^2 * e^x.
Correct Answer: e^x(x^2 + 2x)
Step 1: Identify the function f(x) = x^2 * e^x. This is a product of two functions: x^2 and e^x.
Step 2: Recall the product rule for derivatives. The product rule states that if you have two functions u(x) and v(x), then the derivative of their product is u'v + uv'.
Step 3: Assign u = x^2 and v = e^x. Now we need to find the derivatives of u and 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 8: Factor out e^x from both terms: f'(x) = e^x(2x + x^2).
Step 9: Rearrange the terms inside the parentheses: f'(x) = e^x(x^2 + 2x).
Product Rule – The product rule is used to find the derivative of a product of two functions, stating that if u(x) and v(x) are functions, then (u*v)' = u'v + uv'.
Exponential Functions – Understanding the properties of the exponential function e^x and its derivative, which is e^x.
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