Question

Suppose g, h, and j are differentiable functions with the values for the function and derivative given in the following table:

Use the table to calculate the values of the derivatives listed in Exercises 9–16.

If$f\left(x\right)=g\left(h\right(x\left)\right)$, find$f\text{'}\left(3\right)$

Short answer

The value of$f\text{'}\left(3\right)=-2$

Step-by-step solution

Step 1. Given information

Function is:$f\left(x\right)=g\left(h\right(x\left)\right)$

Given table:

Step 2. Find f'(3) using chain rule:

Since $f\left(x\right)=g\left(h\right(x\left)\right)$

Hence, according to the chain rule of derivative:

$f\text{'}\left(x\right)=g\text{'}\left(h\right(x\left)\right)\times h\text{'}\left(x\right)$

$f\text{'}\left(3\right)=g\text{'}\left(h\right(3\left)\right)\times h\text{'}\left(3\right)$

From the given table we can see that when $x=3$

then $$\begin{gathered} h\left(3\right)=0 \\ h\text{'}\left(3\right)=1 \end{gathered}$$

$g\text{'}\left(0\right)=-2$

Substitute all these value in the above derivative:

$$\begin{gathered} f\text{'}\left(3\right)=-2\times 1 \\ =-2 \end{gathered}$$