Video Transcript
The following table gives values for the functions 𝑓 and 𝑔 and their derivatives for certain values of 𝑥. If the function ℎ of 𝑥 is equal to 𝑓 of 𝑔 of 𝑥, what is the value of ℎ prime of one?
We can see in the table we’ve been given values of 𝑓 of 𝑥, 𝑔 of 𝑥, 𝑓 prime of 𝑥, and 𝑔 prime of 𝑥 for the 𝑥-values zero, one, and two. However, we’ve been asked to find the value of ℎ prime of one, where ℎ of 𝑥 is equal to 𝑓 of 𝑔 of 𝑥. Now, ℎ of 𝑥 is a composite function. It’s a function of a function. We’re taking input value 𝑥, apply the function 𝑔, and then apply the function 𝑓 to the result. So in order to find ℎ prime of one, we need to know how we find the derivative of a composite function.
Well, this is when we would use the chain rule, which tells us that ℎ prime of 𝑥 is equal to 𝑔 prime of 𝑥 multiplied by 𝑓 prime of 𝑔 of 𝑥. That’s the derivative of the inner function multiplied by the derivative of the outer function with the inner function still inside it. However, we don’t want to know the general derivative ℎ prime of 𝑥. We just want to know its value when 𝑥 is equal to one. So we substitute one into our derivative. And we have that ℎ prime of one will be equal to 𝑔 prime of one multiplied by 𝑓 prime of 𝑔 of one.
Now, let’s consider how we can use the table to find some of these values. First, we look in the column 𝑔 prime of 𝑥. And we see that 𝑔 prime of one is equal to six. Next, we look in the column 𝑔 of 𝑥. And we see that 𝑔 of one is equal to zero. So ℎ prime of one is equal to six multiplied by 𝑓 prime of zero. Finally, we look in the column 𝑓 prime of 𝑥. And we see that 𝑓 prime of zero is equal to two. So our value for ℎ prime of one becomes six multiplied by two which is equal to 12.
We’ve therefore found the value of ℎ prime of one. That’s the derivative of the function ℎ of 𝑥 evaluated when 𝑥 is equal to one and seeing that it is equal to 12. We did this by applying the chain rule, which tells us that if ℎ of 𝑥 is a composite function 𝑓 of 𝑔 of 𝑥, then ℎ prime of 𝑥 is equal to 𝑔 prime of 𝑥 multiplied by 𝑓 prime of 𝑔 of 𝑥. That’s the derivative of the inner function multiplied by the derivative of the outer function with the inner function still inside.
