Question

1. if $y = x^{2}e^{x}$, then $\frac{dy}{dx}=$

Explanation:

Step1: Apply the product rule

The product rule states that if \(y = u\cdot v\), then \(y^\prime=u^\prime v + uv^\prime\). Let \(u = x^{2}\) and \(v=e^{x}\). First, find \(u^\prime\) and \(v^\prime\).
\(u^\prime=\frac{d}{dx}(x^{2}) = 2x\) and \(v^\prime=\frac{d}{dx}(e^{x})=e^{x}\)

Step2: Substitute into the product rule

\(\frac{dy}{dx}=u^\prime v+uv^\prime\)
Substitute \(u = x^{2}\), \(u^\prime = 2x\), \(v = e^{x}\), and \(v^\prime=e^{x}\) into the formula:
\(\frac{dy}{dx}=2x\cdot e^{x}+x^{2}\cdot e^{x}\)
Factor out \(x e^{x}\):
\(\frac{dy}{dx}=x e^{x}(2 + x)\)

Answer:

\(x e^{x}(x + 2)\)