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

Short answer

The value of$f\text{'}\left(3\right)=0$

Step-by-step solution

Step 1. Given information

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

Given table:

Step 2. Find f'(3) using chain rule: 

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

Hence, according to the chain rule of derivative:

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

From the given table we can see that

$$\begin{gathered} g\left(3\right)=-3 \\ g\text{'}\left(3\right)=0 \end{gathered}$$

Substitute all these values in the above derivative:

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