Question

use the chart to find ( h(2) ) in simplest form.

h(x)=\frac{f(x)}{g(x)}

Explanation:

Step1: Apply the 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 \(x = 2\) into the quotient - rule formula

We know that \(f(2)=9\), \(f^{\prime}(2)=12\), \(g(2)=3\), and \(g^{\prime}(2)=-7\).
Substitute these values into \(h^{\prime}(x)=\frac{f^{\prime}(x)g(x)-f(x)g^{\prime}(x)}{[g(x)]^{2}}\):

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Step3: Simplify the numerator and denominator

First, calculate the numerator: \(12\times3-9\times(-7)=36 + 63=99\).
The denominator is \(3^{2}=9\).
Then \(h^{\prime}(2)=\frac{99}{9}\)

Step4: Final simplification

\(\frac{99}{9}=11\)

Answer:

\(11\)