Question

section 2.6: chain rule (homework)
score: 120/170 answered: 12/17
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question 13
0/10 pts 4 99 details
find the following using the table below.

x	1	2	3	4
f(x)	1	4	3	2
f(x)	1	4	2	3
g(x)	3	2	4	1
g(x)	4	2	3	1

h(2) if h(x)=f(x)·g(x)
h(2) if h(x)=\frac{f(x)}{g(x)}
h(2) if h(x)=f(g(x))
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Explanation:

Step1: Recall product - rule formula

The product - rule states that if $h(x)=f(x)\cdot g(x)$, then $h^{\prime}(x)=f^{\prime}(x)g(x)+f(x)g^{\prime}(x)$.
Substitute $x = 2$: $h^{\prime}(2)=f^{\prime}(2)g(2)+f(2)g^{\prime}(2)$.
From the table, $f^{\prime}(2) = 4$, $g(2)=2$, $f(2)=4$, $g^{\prime}(2)=2$.
$h^{\prime}(2)=4\times2 + 4\times2=8 + 8=16$.

Step2: Recall quotient - rule formula

The quotient - rule states that if $h(x)=\frac{f(x)}{g(x)}$, then $h^{\prime}(x)=\frac{f^{\prime}(x)g(x)-f(x)g^{\prime}(x)}{g^{2}(x)}$.
Substitute $x = 2$: $h^{\prime}(2)=\frac{f^{\prime}(2)g(2)-f(2)g^{\prime}(2)}{g^{2}(2)}$.
From the table, $f^{\prime}(2) = 4$, $g(2)=2$, $f(2)=4$, $g^{\prime}(2)=2$.
$h^{\prime}(2)=\frac{4\times2-4\times2}{2^{2}}=\frac{8 - 8}{4}=0$.

Step3: Recall chain - rule formula

The chain - rule states that if $h(x)=f(g(x))$, then $h^{\prime}(x)=f^{\prime}(g(x))\cdot g^{\prime}(x)$.
Substitute $x = 2$: First, find $g(2)=2$. Then $h^{\prime}(2)=f^{\prime}(g(2))\cdot g^{\prime}(2)=f^{\prime}(2)\cdot g^{\prime}(2)$.
From the table, $f^{\prime}(2) = 4$ and $g^{\prime}(2)=2$.
$h^{\prime}(2)=4\times2=8$.

Answer:

$h^{\prime}(2)$ for $h(x)=f(x)\cdot g(x)$ is $16$.
$h^{\prime}(2)$ for $h(x)=\frac{f(x)}{g(x)}$ is $0$.
$h^{\prime}(2)$ for $h(x)=f(g(x))$ is $8$.