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Worked example: Chain rule with table

Through a worked example, we explore the Chain rule with a table. Using specific x-values for functions f and g, and their derivatives, we collaboratively evaluate the derivative of a composite function F(x) = f(g(x)). By applying the chain rule, we illuminate the process, making it easy to understand.

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Video transcript

- [Voiceover] The following table lists the values of functions f and g and of their derivatives, f-prime and g-prime for the x values negative two and four. And so you can see for x equals negative two, x equals four, they gave us the values of f, g, f-prime, and g-prime. Let function capital-F be defined as the composition of f and g. It's lowercase-f of g of x, and they want us to evaluate f-prime of four. So you might immediately recognize that if I have a function that can be viewed as the composition of other functions that the chain rule will apply here. And so, and I'm just gonna restate the chain rule, the derivative of capital-F is going to be the derivative of lowercase-f, the outside function with respect to the inside function. So lowercase-F-prime of g of x times the derivative of the inside function with respect to x times g-prime of x. And if we're looking for F-prime of four, F-prime of four, well everywhere we see an x we replace it with a four. That's gonna be lowercase-f-prime of g of four times g-prime of four. Now how do we figure this out? They haven't given us explicitly the values of the functions for all xs, but they've given it to us at some interesting points. So the first thing you might wanna figure out is well what is g of four going to be? Well they tell us: when x is equal to four, g of four is negative two. This tells us that the value of g of x takes on when x is equal to four is negative two. So this right over here is negative two. And so this first part is f-prime of negative two. So what is f-prime, what is f-prime of negative two? Well when x is equal to negative two, f-prime is equal to one. So this right over here is f-prime of negative two. That is equal to one. And now we just have to figure out what g-prime of four is. Well when, let me circle this, g-prime of four, when x is equal to four, and I'll scroll down a little bit, when x is equal to four, g-prime takes on the value eight. So there you have it. F-prime of four is equal to one times eight which is equal to eight, and we're done.