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• - [Instructor] Let's say that we have the function g of x,

• and it is equal to the definite integral from 19 to x

• of the cube root of t dt.

• And what I'm curious about finding

• or trying to figure out is, what is g prime of 27?

• What is that equal to?

• Pause this video and try to think about it,

• and I'll give you a little bit of a hint.

• Think about the second fundamental theorem of calculus.

• All right, now let's work on this together.

• So we wanna figure out what g prime,

• we could try to figure out what g prime of x is,

• and then evaluate that at 27,

• and the best way that I can think about doing that

• is by taking the derivative of both sides of this equation.

• So let's take the derivative of both sides of that equation.

• So the left-hand side, we'll take the derivative

• with respect to x of g of x,

• and the right-hand side, the derivative with respect to x

• of all of this business.

• Now, the left-hand side is pretty straight forward.

• The derivative with respect to x of g of x,

• that's just going to be g prime of x,

• but what is the right-hand side going to be equal to?

• Well, that's where the second fundamental theorem

• of calculus is useful.

• I'll write it right over here.

• Second fundamental, I'll abbreviate a little bit,

• theorem of calculus.

• It tells us, let's say we have some function capital F of x,

• and it's equal to the definite integral from a,

• sum constant a to x of lowercase f of t dt.

• The second fundamental theorem of calculus tells us

• that if our lowercase f,

• if lowercase f is continuous on the interval from a to x,

• so I'll write it this way,

• on the closed interval from a to x,

• then the derivative of our capital f of x,

• so capital F prime of x is just going to be equal

• to our inner function f evaluated at x instead of t

• is going to become lowercase f of x.

• Now, I know when you first saw this,

• you thought that, "Hey, this might be some cryptic thing

• "that you might not use too often."

• Well, we're gonna see that it's actually very, very useful

• and even in the future, and some of you might already know,

• there's multiple ways to try

• to think about a definite integral like this,

• and you'll learn it in the future.

• But this can be extremely simplifying,

• especially if you have a hairy definite integral like this,

• and so this just tells us, hey, look, the derivative

• with respect to x of all of this business,

• first we have to check that our inner function,

• which would be analogous to our lowercase f here,

• is this continuous on the interval from 19 to x?

• Well, no matter what x is,

• this is going to be continuous over that interval,

• because this is continuous for all x's,

• and so we meet this first condition or our major condition,

• and so then we can just say, all right,

• then the derivative of all of this

• is just going to be this inner function replacing t with x.

• So we're going to get the cube root,

• instead of the cube root of t,

• you're gonna get the cube root of x.

• And so we can go back to our original question,

• what is g prime of 27 going to be equal to?

• Well, it's going to be equal to the cube root of 27,

• which is of course equal to three, and we're done.

- [Instructor] Let's say that we have the function g of x,

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# 微積分の基本定理で微分を見つける｜AP®︎ 微積分AB｜カーンアカデミー (Finding derivative with fundamental theorem of calculus | AP®︎ Calculus AB | Khan Academy)

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林宜悉 に公開 2021 年 01 月 14 日