Question

the table above gives values of the differentiable functions ( f ) and ( g ) and of their derivatives ( f ) and ( g ), at selected values of ( x ). if ( h(x)=f(g(x)) ), what is the slope of the graph of ( h ) at ( x = 2 )?

Explanation:

Step1: Apply the 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 \(x = 2\)

We need to find \(h^{\prime}(2)\). First, find \(g(2)\) from the table. When \(x = 2\), \(g(2)=-1\). Then \(h^{\prime}(2)=f^{\prime}(g(2))\cdot g^{\prime}(2)\).
Since \(g(2)=-1\), we substitute into \(f^{\prime}(g(2))\) and \(g^{\prime}(2)\). From the table, when \(x=-1\), \(f^{\prime}(-1) = 3\) and when \(x = 2\), \(g^{\prime}(2)=2\).
So \(h^{\prime}(2)=f^{\prime}(g(2))\cdot g^{\prime}(2)=f^{\prime}(-1)\cdot g^{\prime}(2)\).

Step3: Calculate the value

Substitute \(f^{\prime}(-1) = 3\) and \(g^{\prime}(2)=2\) into the expression. \(h^{\prime}(2)=3\times2=6\).

Answer:

\(6\)