let h(x)=f(g(x)) and p(x)=g(f(x)). use the table to compute the following derivatives. a. h(1) b. p(3) x…

Explanation:

Step1: Recall the chain - rule

The chain - rule states that if $h(x)=f(g(x))$, then $h^{\prime}(x)=f^{\prime}(g(x))\cdot g^{\prime}(x)$ and if $p(x)=g(f(x))$, then $p^{\prime}(x)=g^{\prime}(f(x))\cdot f^{\prime}(x)$.

Step2: Compute $h^{\prime}(1)$

First, find $g(1)$ from the table. When $x = 1$, $g(1)=2$. Then, $h^{\prime}(1)=f^{\prime}(g(1))\cdot g^{\prime}(1)$. Since $g(1)=2$ and from the table $g^{\prime}(1)=3$ and $f^{\prime}(2)= - 2$, then $h^{\prime}(1)=f^{\prime}(2)\cdot g^{\prime}(1)=(-2)\times3=-6$.

Step3: Compute $p^{\prime}(3)$

First, find $f(3)$ from the table. When $x = 3$, $f(3)=4$. Then, $p^{\prime}(3)=g^{\prime}(f(3))\cdot f^{\prime}(3)$. Since $f(3)=4$ and from the table $f^{\prime}(3)= - 7$ and $g^{\prime}(4)=5$, then $p^{\prime}(3)=g^{\prime}(4)\cdot f^{\prime}(3)=5\times(-7)=-35$.

Answer:

$h^{\prime}(1)=-6$ $p^{\prime}(3)=-35$