Question

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

Explanation:

Step1: Find first - derivative using chain - rule

Let $u = 2x^{3}$, then $y = e^{u}$. The chain - rule states that $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. We know that $\frac{dy}{du}=e^{u}$ and $\frac{du}{dx}=6x^{2}$. So, $\frac{dy}{dx}=e^{2x^{3}}\cdot6x^{2}=6x^{2}e^{2x^{3}}$.

Step2: Find second - derivative using product - rule

The product - rule states that if $y = uv$, where $u = 6x^{2}$ and $v = e^{2x^{3}}$, then $\frac{d^{2}y}{dx^{2}}=u\frac{dv}{dx}+v\frac{du}{dx}$.
First, find $\frac{dv}{dx}$: Let $t = 2x^{3}$, then $v = e^{t}$, and $\frac{dv}{dt}=e^{t}$, $\frac{dt}{dx}=6x^{2}$, so $\frac{dv}{dx}=e^{2x^{3}}\cdot6x^{2}$. Also, $\frac{du}{dx}=12x$.
Then $\frac{d^{2}y}{dx^{2}}=6x^{2}\cdot6x^{2}e^{2x^{3}}+e^{2x^{3}}\cdot12x$.

Step3: Factor the result

Factor out $6xe^{2x^{3}}$ from the expression: $\frac{d^{2}y}{dx^{2}}=6xe^{2x^{3}}(6x^{3} + 2)$.

Answer:

$6xe^{2x^{3}}(6x^{3}+2)$