Question

let h(x)=f(g(x)) and p(x)=g(f(x)). use the table to compute the following derivatives.
a. h(4)
b. p(2)
\begin{array}{|c|c|c|c|c|}hline x&1&2&3&4\hline f(x)&4&3&2&1\hline f(x)& - 7& - 8& - 1& - 9\hline g(x)&1&4&3&2\hline g(x)&\frac{4}{9}&\frac{1}{9}&\frac{7}{9}&\frac{2}{9}\hlineend{array}
h(4)=\square (simplify your answer)

Explanation:

Step1: Apply chain - rule for $h(x)$

The chain - rule states that if $h(x)=f(g(x))$, then $h^{\prime}(x)=f^{\prime}(g(x))\cdot g^{\prime}(x)$. To find $h^{\prime}(4)$, we substitute $x = 4$ into the formula: $h^{\prime}(4)=f^{\prime}(g(4))\cdot g^{\prime}(4)$.

Step2: Find $g(4)$ from the table

From the table, when $x = 4$, $g(4)=2$.

Step3: Find $f^{\prime}(g(4))$

Since $g(4)=2$, we need to find $f^{\prime}(2)$. From the table, $f^{\prime}(2)=-8$.

Step4: Find $g^{\prime}(4)$

From the table, when $x = 4$, $g^{\prime}(4)=\frac{2}{9}$.

Step5: Calculate $h^{\prime}(4)$

$h^{\prime}(4)=f^{\prime}(g(4))\cdot g^{\prime}(4)=-8\times\frac{2}{9}=-\frac{16}{9}$.

for $p^{\prime}(2)$

Step1: Apply chain - rule for $p(x)$

The chain - rule states that if $p(x)=g(f(x))$, then $p^{\prime}(x)=g^{\prime}(f(x))\cdot f^{\prime}(x)$. To find $p^{\prime}(2)$, we substitute $x = 2$ into the formula: $p^{\prime}(2)=g^{\prime}(f(2))\cdot f^{\prime}(2)$.

Step2: Find $f(2)$ from the table

From the table, when $x = 2$, $f(2)=3$.

Step3: Find $g^{\prime}(f(2))$

Since $f(2)=3$, we need to find $g^{\prime}(3)$. From the table, $g^{\prime}(3)=\frac{7}{9}$.

Step4: Find $f^{\prime}(2)$

From the table, when $x = 2$, $f^{\prime}(2)=-8$.

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

$p^{\prime}(2)=g^{\prime}(f(2))\cdot f^{\prime}(2)=\frac{7}{9}\times(-8)=-\frac{56}{9}$.

Answer:

$-\frac{16}{9}$