the table above gives values of the differentiable functions ( f ) and ( g ) and of their derivatives ( f )…

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 )?

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 )?

Answer

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)