Let's find the value of [tex]\((f \circ g)^{\prime}(3)\)[/tex] using the given table.
We need to evaluate the derivative of the composition of two functions, [tex]\(f\)[/tex] and [tex]\(g\)[/tex], at [tex]\(x = 3\)[/tex]. The chain rule tells us that:
[tex]\[
(f \circ g)'(x) = f'(g(x)) \cdot g'(x)
\][/tex]
First, we need to determine several values from the table for [tex]\(x = 3\)[/tex]:
1. [tex]\( g(3) \)[/tex]: According to the table, [tex]\( g(3) = 11 \)[/tex].
2. [tex]\( g'(3) \)[/tex]: According to the table, [tex]\( g'(3) = 8 \)[/tex].
3. Next, we need [tex]\( f'(g(3)) \)[/tex]. We already determined that [tex]\( g(3) = 11 \)[/tex], so we need [tex]\( f'(11) \)[/tex]:
[tex]\[
f'(11) = -6
\][/tex]
Now, applying the chain rule:
[tex]\[
(f \circ g)'(3) = f'(g(3)) \cdot g'(3) = f'(11) \cdot g'(3) = -6 \cdot 8 = -48
\][/tex]
Thus, the value of [tex]\((f \circ g)^{\prime}(3)\)[/tex] is:
[tex]\[
-48
\][/tex]
