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 $f\left(x\right)={\left(g\left(x\right)\right)}^3$, find$f\text{'}(-2)$

Short answer

The value of$f\text{'}(-2)=6$

Step-by-step solution

Step 1. Given information:

Function is: $f\left(x\right)={\left(g\left(x\right)\right)}^3$

Given table:

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

Since $f\left(x\right)={\left(g\left(x\right)\right)}^3$,

Hence, according to the chain rule of derivative:

role="math" localid="1648385862063" $$\begin{gathered} f\text{'}\left(x\right)=3{\left(g\left(x\right)\right)}^2\times g\text{'}\left(x\right) \\ f\text{'}(-2)=3{\left(g(-2)\right)}^2\times g\text{'}(-2) \end{gathered}$$

From the given table we can see that

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

Substitute all these values in the above derivative:

$$\begin{gathered} f\text{'}(-2)=3{\left(-1\right)}^2\times 2 \\ =3\times 1\times 2 \\ =6 \end{gathered}$$