Question

use the chart to find $h(4)$ in simplest form.
$h(x)=f(g(x))$

Explanation:

Step1: Apply chain rule

The chain rule states that if \(h(x)=f(g(x))\), then \(h^{\prime}(x)=f^{\prime}(g(x))\cdot g^{\prime}(x)\).

Step2: Substitute \(x = 4\)

When \(x = 4\), first find \(g(4)=6\). Then \(h^{\prime}(4)=f^{\prime}(g(4))\cdot g^{\prime}(4)\).
Since \(g(4) = 6\), we substitute into \(f^{\prime}\) and \(g^{\prime}\). So \(h^{\prime}(4)=f^{\prime}(6)\cdot g^{\prime}(4)\).

Step3: Use table values

From the table, \(f^{\prime}(6)=- 11\) and \(g^{\prime}(4)=-3\). Then \(h^{\prime}(4)=(-11)\times(-3)\).

Answer:

\(33\)