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Chain rule
Worked example: Chain rule with table
AP.CALC:
FUN‑3 (EU)
, FUN‑3.C (LO)
, FUN‑3.C.1 (EK)
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.