Question: What is the derivative of f(x) = e^x * x^2?
Options:
e^x * (2x + 1)
e^x * (x^2 + 2)
e^x * 2x
2e^x * x
Correct Answer: e^x * (2x + 1)
Solution:
Using the product rule, f\'(x) = e^x * x^2 + e^x * 2x = e^x * (x^2 + 2x).
What is the derivative of f(x) = e^x * x^2?
Practice Questions
Q1
What is the derivative of f(x) = e^x * x^2?
e^x * (2x + 1)
e^x * (x^2 + 2)
e^x * 2x
2e^x * x
Questions & Step-by-Step Solutions
What is the derivative of f(x) = e^x * x^2?
Step 1: Identify the function f(x) = e^x * x^2. This is a product of two functions: e^x and x^2.
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) = e^x and v(x) = x^2. Now we need to find the derivatives of u and v.
Step 4: Calculate the derivative of u(x). The derivative of e^x is e^x, so u' = e^x.
Step 5: Calculate the derivative of v(x). The derivative of x^2 is 2x, so v' = 2x.
Step 6: Apply the product rule. Substitute u, v, u', and v' into the product rule formula: f'(x) = u'v + uv'.
Step 7: Substitute the values: f'(x) = (e^x)(x^2) + (e^x)(2x).
Step 8: Factor out e^x from both terms: f'(x) = e^x * (x^2 + 2x).
Step 9: Write the final answer: The derivative of f(x) = e^x * x^2 is f'(x) = e^x * (x^2 + 2x).
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