Q. 16 - 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  $f\left(x\right)=h\left(h\left(h\left(x\right)\right)\right)$, find $f\text{'}\left(1\right)$

Step-by-Step Solution

Answer

The value of $f\text{'}\left(1\right)=12$

Step 1. Given information:

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

Given table:

Step 2. Find f ' ( 1 ) using chain rule:

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

Hence, according to the chain rule of derivative:

$$\begin{gathered} f\text{'}\left(x\right)=h\text{'}\left(h\left(h\left(x\right)\right)\right)\times h\text{'}\left(h\left(x\right)\right)\times h\text{'}\left(x\right) \\ f\text{'}\left(1\right)=h\text{'}\left(h\left(h\left(1\right)\right)\right)\times h\text{'}\left(h\left(1\right)\right)\times h\text{'}\left(1\right) \end{gathered}$$

From the given table we can see that

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

Substitute all these values in the above derivative:

$f\text{'}\left(1\right)=h\text{'}\left(h\left(-1\right)\right)\times h\text{'}\left(-1\right)\times \left(-2\right)$

From table,

$$\begin{gathered} h(-1)=0 \\ h\text{'}(-1)=-2 \end{gathered}$$

$$\begin{gathered} f\text{'}\left(1\right)=h\text{'}\left(0\right)\times \left(-2\right)\times \left(-2\right) \\ f\text{'}\left(1\right)=4\times h\text{'}\left(0\right) \end{gathered}$$

In table, $h\text{'}\left(0\right)=3$

so,$$\begin{gathered} f\text{'}\left(1\right)=4\times 3 \\ =12 \end{gathered}$$