Question

for the function y=(4x + 3)(2x^2+7x - 1), find $\frac{d^{2}y}{dx^{2}}$. answer: $\frac{d^{2}y}{dx^{2}}=$

Explanation:

Step1: Expand the function

\[

$$\begin{align*} y&=(4x + 3)(2x^{2}+7x - 1)\\ &=4x(2x^{2}+7x - 1)+3(2x^{2}+7x - 1)\\ &=8x^{3}+28x^{2}-4x + 6x^{2}+21x - 3\\ &=8x^{3}+34x^{2}+17x - 3 \end{align*}$$

\]

Step2: Find the first - derivative

Using the power rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$, we have:
\[

$$\begin{align*} \frac{dy}{dx}&=\frac{d}{dx}(8x^{3}+34x^{2}+17x - 3)\\ &=8\times3x^{2}+34\times2x+17\\ &=24x^{2}+68x + 17 \end{align*}$$

\]

Step3: Find the second - derivative

Again, using the power rule:
\[

$$\begin{align*} \frac{d^{2}y}{dx^{2}}&=\frac{d}{dx}(24x^{2}+68x + 17)\\ &=24\times2x+68\\ &=48x + 68 \end{align*}$$

\]

Answer:

$48x + 68$