Question

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 | 1 | 2 | 3 | 4 | f(x) | 4 | 1 | 2 | 3 | f(x) | - 4 | - 6 | - 3 | - 7 | g(x) | 1 | 2 | 3 | 4 | g(x) | 1/5 | 3/5 | 2/5 | 4/5 | h(1)= (simplify your answer.)

Explanation:

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

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

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

First, find $g(1)$ from the table. When $x = 1$, $g(1)=1$. Then find $f^{\prime}(g(1))$ and $g^{\prime}(1)$. Since $g(1)=1$, $f^{\prime}(g(1))=f^{\prime}(1)=-4$. And $g^{\prime}(1)=\frac{1}{5}$. So $h^{\prime}(1)=f^{\prime}(g(1))\cdot g^{\prime}(1)=-4\times\frac{1}{5}=-\frac{4}{5}$.

Answer:

$-\frac{4}{5}$