use the chart to find h(1) in simplest form.\n| x | f(x) | f(x) | g(x) | g(x) |\n| 1 | 8 | -2 | 2 | -8…

Explanation:

Step1: Recall quotient - rule

The quotient - rule states that if $h(x)=\frac{f(x)}{g(x)}$, then $h^{\prime}(x)=\frac{f^{\prime}(x)g(x)-f(x)g^{\prime}(x)}{[g(x)]^{2}}$.

Step2: Substitute $x = 1$

We know that $f(1) = 8$, $f^{\prime}(1)=-2$, $g(1)=2$, and $g^{\prime}(1)=-8$. Substitute these values into the quotient - rule formula: [ \begin{align*} h^{\prime}(1)&=\frac{f^{\prime}(1)g(1)-f(1)g^{\prime}(1)}{[g(1)]^{2}}\ &=\frac{(-2)\times2-8\times(-8)}{2^{2}} \end{align*} ]

Step3: Calculate the numerator and denominator

First, calculate the numerator: $(-2)\times2-8\times(-8)=-4 + 64=60$. The denominator is $2^{2}=4$.

Step4: Find the value of $h^{\prime}(1)$

$h^{\prime}(1)=\frac{60}{4}=15$.

Answer:

$15$