Video Transcript
Two functions, 𝑓 of 𝑥 and 𝑔 of 𝑥, are continuous and differentiable for all real numbers. Some values of the functions and their derivatives are given in the table. Based on the given table, what is the value of ℎ prime of three if ℎ of 𝑥 equals 𝑔 of 𝑓 of 𝑥.?
Let’s consider this function ℎ of 𝑥, first of all. It’s equal to 𝑔 of 𝑓 of 𝑥, which means it is a composite function, a function of a function. We take an input 𝑥, apply the function 𝑓, and then we apply the function 𝑔 to the result. We’re asked to work out the value of ℎ prime of three. Which means, we need to work out the derivative of this composite functions ℎ of 𝑥. To find the derivative of a composite function, we can apply the chain rule, which tells us that ℎ prime of 𝑥 will be equal to 𝑓 prime of 𝑥, that’s the derivative of the inner function, multiplied by 𝑔 prime of 𝑓 of 𝑥. That’s the derivative of the outer function, with the inner function still inside.
However, we don’t need to know what this derivative is for any value of 𝑥. We want to evaluate it specifically when 𝑥 is equal to three. So we know the ℎ prime of three will be equal to 𝑓 prime of three multiplied by 𝑔 prime of 𝑓 of three. Let’s consider how we can use the table in order to work this out. First, we see in the table that the value of 𝑓 prime of three is six. Next, we see that the value of 𝑓 of three is four. So we have six 𝑔 prime of four. That’s the derivative of 𝑓 evaluated at three multiplied by the derivative of 𝑔 evaluated at 𝑓 of three.
We then see further in the table that 𝑔 prime of four is equal to eight. So the derivative, ℎ prime of three, become six multiplied by eight. Six multiplied by eight is 48. So we’ve found ℎ prime of three.
In this question, we applied the chain rule which tells us how to find the derivative of a composite function. If ℎ of 𝑥 is equal to 𝑔 of 𝑓 of 𝑥, then ℎ prime of 𝑥 is equal to 𝑓 prime of 𝑥 multiplied by 𝑔 prime of 𝑓 of 𝑥. That’s the derivative of the inner function multiplied by the derivative of the outer function, with the inner function still inside.
