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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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用微積分基本定理求導數 | 金剛石微積分AB︎ | 可汗學院 (Finding derivative with fundamental theorem of calculus | AP®︎ Calculus AB | Khan Academy)

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    林宜悉 發佈於 2021 年 01 月 14 日
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