Question

the following table lists the values of functions g and h, and of their derivatives, g and h, for the x-values -3 and 2.

x	g(x)	h(x)	g(x)	h(x)
2	4	-3	3	-4

evaluate \\(\frac{d}{dx}\leftg\left(h(x)\
ight)\
ight\\) at \\(x = 2\\).

Explanation:

Step1: Apply Chain Rule

The chain rule for differentiation states that if we have a composite function \( y = g(h(x)) \), then its derivative is \( \frac{dy}{dx}=g'(h(x))\cdot h'(x) \). So we need to evaluate \( g'(h(2))\cdot h'(2) \).

Step2: Find \( h(2) \)

From the table, when \( x = 2 \), \( h(2)=- 3 \).

Step3: Find \( g'(h(2)) \)

Since \( h(2)=-3 \), we need to find \( g'(-3) \). From the table, when \( x=-3 \), \( g'(-3) = - 7 \).

Step4: Find \( h'(2) \)

From the table, when \( x = 2 \), \( h'(2)=-4 \).

Step5: Multiply the results

Now we multiply \( g'(h(2)) \) and \( h'(2) \). So \( g'(h(2))\cdot h'(2)=g'(-3)\cdot h'(2)=(-7)\times(-4) = 28 \).

Answer:

\( 28 \)