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• - [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.

- [Voiceover] Let f be a function such that

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# Worked example: Quotient rule with table | Derivative rules | AP Calculus AB | Khan Academy

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yukang920108 posted on 2022/07/12
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