Question

for the function y = 3xe^x, find $\frac{d^{2}y}{dx^{2}}$. simplify and factor. answer: $\frac{d^{2}y}{dx^{2}}=$

Explanation:

Step1: Find the first - derivative

The product rule states that if $y = uv$, where $u$ and $v$ are functions of $x$, then $y^\prime=u^\prime v + uv^\prime$. For $y = 3xe^{x}$, let $u = 3x$ and $v = e^{x}$. Then $u^\prime=3$ and $v^\prime = e^{x}$. So, $\frac{dy}{dx}=3e^{x}+3xe^{x}=3e^{x}(1 + x)$.

Step2: Find the second - derivative

Again, use the product rule on $\frac{dy}{dx}=3e^{x}(1 + x)$. Let $u = 3(1 + x)$ and $v = e^{x}$. Then $u^\prime=3$ and $v^\prime = e^{x}$. So, $\frac{d^{2}y}{dx^{2}}=3e^{x}+3(1 + x)e^{x}$.

Step3: Simplify and factor

Factor out $3e^{x}$ from the right - hand side: $\frac{d^{2}y}{dx^{2}}=3e^{x}(1+(1 + x))=3e^{x}(x + 2)$.

Answer:

$3e^{x}(x + 2)$