Q. 9 - Calculus

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 $f\text{'}\left(3\right)$

Step-by-Step Solution

Answer

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

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:

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