Question

use the product rule to find the derivative. y=(3x^{2}+2)(4x - 3) y=□

Explanation:

Step1: Recall product - rule

The product rule states that if $y = u\cdot v$, then $y'=u'v + uv'$. Let $u = 3x^{2}+2$ and $v = 4x - 3$.

Step2: Find $u'$

Differentiate $u = 3x^{2}+2$ with respect to $x$. Using the power - rule $\frac{d}{dx}(ax^{n})=nax^{n - 1}$, we have $u'=\frac{d}{dx}(3x^{2}+2)=6x$.

Step3: Find $v'$

Differentiate $v = 4x - 3$ with respect to $x$. Using the power - rule, $v'=\frac{d}{dx}(4x - 3)=4$.

Step4: Apply product - rule

Substitute $u$, $u'$, $v$, and $v'$ into the product rule formula:
$y'=u'v+uv'=(6x)(4x - 3)+(3x^{2}+2)\times4$.
Expand the expression:
\[

$$\begin{align*} y'&=24x^{2}-18x + 12x^{2}+8\\ &=(24x^{2}+12x^{2})-18x + 8\\ &=36x^{2}-18x + 8 \end{align*}$$

\]

Answer:

$36x^{2}-18x + 8$