Question

a. find the derivative of g(x)=(8x - 1)(3x + 7)
g(x)=
b. what is the slope at x = - 2?
slope at x = - 2:

Explanation:

Step1: Apply product - rule

The product - rule states that if $g(x)=u(x)v(x)$, then $g^{\prime}(x)=u^{\prime}(x)v(x)+u(x)v^{\prime}(x)$. Let $u(x)=8x - 1$ and $v(x)=3x + 7$. Then $u^{\prime}(x)=8$ and $v^{\prime}(x)=3$.
$g^{\prime}(x)=8(3x + 7)+(8x - 1)\times3$

Step2: Expand and simplify

Expand the expression:
\[

$$\begin{align*} g^{\prime}(x)&=24x+56 + 24x-3\\ &=(24x + 24x)+(56 - 3)\\ &=48x + 53 \end{align*}$$

\]

Step3: Find the slope at $x = - 2$

Substitute $x=-2$ into $g^{\prime}(x)$.
$g^{\prime}(-2)=48\times(-2)+53$
$g^{\prime}(-2)=-96 + 53=-43$

Answer:

a. $g^{\prime}(x)=48x + 53$
b. Slope at $x=-2$: $-43$