Question

1. derivatives using tables let $h(x)=f(g(x))$ and $p(x)=g(f(x))$. use the table to compute the following derivatives. a. $h(3)$ b. $h(2)$ c. $p(4)$ d. $p(2)$ e. $h(5)$

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 $h^{\prime}(3)$

For $h(x) = f(g(x))$, $h^{\prime}(3)=f^{\prime}(g(3))\cdot g^{\prime}(3)$. First, find $g(3)$ from the table, then $f^{\prime}(g(3))$, and $g^{\prime}(3)$ and multiply them.

Step3: Compute $h^{\prime}(2)$

$h^{\prime}(2)=f^{\prime}(g(2))\cdot g^{\prime}(2)$. Find $g(2)$ from the table, then $f^{\prime}(g(2))$, and $g^{\prime}(2)$ and multiply them.

Step4: Compute $p^{\prime}(4)$

For $p(x)=g(f(x))$, $p^{\prime}(4)=g^{\prime}(f(4))\cdot f^{\prime}(4)$. Find $f(4)$ from the table, then $g^{\prime}(f(4))$, and $f^{\prime}(4)$ and multiply them.

Step5: Compute $p^{\prime}(2)$

$p^{\prime}(2)=g^{\prime}(f(2))\cdot f^{\prime}(2)$. Find $f(2)$ from the table, then $g^{\prime}(f(2))$, and $f^{\prime}(2)$ and multiply them.

Step6: Compute $h^{\prime}(5)$

$h^{\prime}(5)=f^{\prime}(g(5))\cdot g^{\prime}(5)$. Find $g(5)$ from the table, then $f^{\prime}(g(5))$, and $g^{\prime}(5)$ and multiply them.

Since the table is not provided, we cannot give the numerical answers. But the general procedure for each part is as described above. If we assume the values from a sample table (not given here):
Let's say $g(3) = a$, $f^{\prime}(a)=b$, $g^{\prime}(3)=c$, then $h^{\prime}(3)=b\cdot c$. And so on for other parts.

Answer:

Without the table, numerical answers cannot be provided. The general forms are:
a. $h^{\prime}(3)=f^{\prime}(g(3))\cdot g^{\prime}(3)$
b. $h^{\prime}(2)=f^{\prime}(g(2))\cdot g^{\prime}(2)$
c. $p^{\prime}(4)=g^{\prime}(f(4))\cdot f^{\prime}(4)$
d. $p^{\prime}(2)=g^{\prime}(f(2))\cdot f^{\prime}(2)$
e. $h^{\prime}(5)=f^{\prime}(g(5))\cdot g^{\prime}(5)$