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