Question

use the chart to find $h(3)$ in simplest form.

$x$	$f(x)$	$f(x)$	$g(x)$	$g(x)$

$h(x)=\frac{f(x)}{g(x)}$
answer attempt 1 out of 2
$h(3)=$

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 $x = 3$

We substitute $x = 3$ into the quotient - rule formula. Given $f(3)=3$, $f^{\prime}(3)=4$, $g(3)=5$, and $g^{\prime}(3)=-6$.
$h^{\prime}(3)=\frac{f^{\prime}(3)g(3)-f(3)g^{\prime}(3)}{[g(3)]^{2}}$.

Step3: Calculate the numerator

$f^{\prime}(3)g(3)-f(3)g^{\prime}(3)=4\times5 - 3\times(-6)=20 + 18=38$.

Step4: Calculate the denominator

$[g(3)]^{2}=5^{2}=25$.

Step5: Find $h^{\prime}(3)$

$h^{\prime}(3)=\frac{38}{25}$.

Answer:

$\frac{38}{25}$