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!