For the following exercises, find f ′(x) for each function f(x) = x2 e^x

To find the derivative of the function \( f(x) = x^2 e^x \), we need to apply the **product rule** because it is the product of two functions: \( x^2 \) and \( e^x \).

### Product Rule:
If \( f(x) = u(x)v(x) \), then:
\[
f'(x) = u'(x)v(x) + u(x)v'(x)
\]
Here, \( u(x) = x^2 \) and \( v(x) = e^x \).

#### Step 1: Differentiate \( u(x) = x^2 \)
\[
u'(x) = 2x
\]

#### Step 2: Differentiate \( v(x) = e^x \)
\[
v'(x) = e^x
\]

#### Step 3: Apply the product rule
\[
f'(x) = u'(x)v(x) + u(x)v'(x)
\]
Substitute \( u'(x), u(x), v'(x), \) and \( v(x) \) into the equation:
\[
f'(x) = (2x)(e^x) + (x^2)(e^x)
\]

#### Step 4: Factor out \( e^x \)
\[
f'(x) = e^x(2x + x^2)
\]

Thus, the derivative is:
\[
f'(x) = e^x(x^2 + 2x)
\]

Would you like more details or have any questions on this process?

Here are 5 related questions to practice:
1. Differentiate \( f(x) = x^3 e^x \).
2. Find the derivative of \( f(x) = x^2 \ln(x) \).
3. Use the product rule to differentiate \( f(x) = x^2 \sin(x) \).
4. Differentiate \( f(x) = (x^2 + 1)e^x \).
5. Apply the product rule to \( f(x) = x^3 \cos(x) \).

**Tip**: Always check if your function is a product of two functions, so you can apply the product rule efficiently!

For the following exercises, find f ′(x) for each function f(x) = x^2 e^x

Learn how to find the derivative of the function f(x) = x^2 e^x using the product rule. This step-by-step guide covers calculus concepts such as derivatives and the product rule. Ideal for students in Grades 11-12 preparing for calculus exams.

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