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AP Calc: FUN‑3 (EU), FUN‑3.B (LO), FUN‑3.B.2 (EK)

- [Voiceover] Let f be
a function such that f of negative one is equal to three, f prime of negative one is equal to five. Let g be the function g of x is equal to two x to the third power. Let capital F be a function defined as, so capital F is defined
as lowercase f of x divided by lowercase g
of x, and they want us to evaluate the derivative of capital F at x equals negative one. So the way that we can do that is, let's just take the
derivative of capital F, and then evaluate it at x equals one. And the way they've set up capital F, this function definition, we can see that it is a quotient of two functions. So if we want to take it's derivative, you might say, well, maybe the quotient rule is important here. And I'll always give you my aside. The quotient rule, I'm
gonna state it right now, it could be useful to know it, but in case you ever forget it, you can derive it pretty quickly from the product rule, and if you know it, the chain rule combined, you can get the quotient rule pretty quick. But let me just state the
quotient rule right now. So if you have some function defined as some function in the numerator divided by some function
in the denominator, we can say its derivative, and this is really just a restatement
of the quotient rule, its derivative is going to
be the derivative of the function of the numerator, so d, dx, f of x, times the function
in the denominator, so times g of x, minus the
function in the numerator, minus f of x, not taking its derivative, times the derivative in the
function of the denominator, d, dx, g of x, all of that over, so all of this is going to be over the function in the denominator squared. So this g of x squared,
g of x, g of x squared. And you can use different
types of notation here. You could say, instead
of writing this with a derivative operator,
you could say this is the same thing as g
prime of x, and likewise, you could say, well that is
the same thing as f prime of x. And so now we just want
to evaluate this thing, and you might say, wait, how
do I evaluate this thing? Well, let's just try it. Let's just say we want to evaluate F prime when x is equal to negative one. So we can write F prime of
negative one is equal to, well everywhere we see an x,
let's put a negative one here. It's going to be f prime of negative one, lowercase f prime, that's
a little confusing, lowercase f prime of negative
one times g of negative one, g of negative one minus f of negative one times g prime of negative one. All of that over, we'll
do that in the same color, so take my color seriously. Alright, all of that over
g of negative one squared. Now can we figure out what
F prime of negative one f of negative one, g of negative one, and g prime of negative
one, what they are? Well some of them, they tell us outright. They tell us f and f
prime at negative one, and for g, we can
actually solve for those. So, let's see, if this is, let's first evaluate g of negative one. G of negative one is going to be two times negative one to the third power. Well negative one to the third
power is just negative one, times two, so this is negative two, and g prime of x, I'll
do it here, g prime of x. Let's use the power rule,
bring that three out front, three times two is six, x,
decrement that exponent, three minus one is two, and
so g prime of negative one is equal to six times
negative one squared. Well negative one squared is just one, so this is going to be equal to six. So we actually know what
all of these values are now. We know, so first we wanna figure out f prime of negative one. Well they tell us that right over here. F prime of negative one is equal to five. So that is five. G of negative one, well we
figured that right here. G of negative one is negative two. So this is negative two. F of negative one, so f of negative one, they tell us that right over there. That is equal to three. And then g prime of negative one, just circle it in this green color, g prime of negative
one, we figured it out. It is equal to six. So this is equal to six. And then finally, g of
negative one right over here. We already figured that out. That was equal to negative two. So this is all going to simplify to... So you have five times negative two, which is negative 10,
minus three times six, which is 18, all of that
over negative two squared. Well negative two squared is
just going to be positive four. So this is going to be equal to negative 28 over positive four, which is equal to negative seven. And there you have it. It looks intimidating at first, but just say, okay, look. I can use the quotient
rule right over here, and then once I apply the quotient rule, I can actually just directly figure out what g of negative one,
g prime of negative one, and they gave us f of negative one, f prime of negative one, so
hopefully you find that helpful.

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