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  • - [Instructor] What we have right over here

  • is the graph of Y is equal to E to the X

  • and what we're going to know by the end of this video

  • is one of the most fascinating ideas in calculus

  • and once again it reinforces the idea

  • that E is really this somewhat magical number.

  • So we're gonna do a little bit of an exploration.

  • Let's just pick some points on this curve

  • of Y is equal to E to the X

  • and think about what the slope of the tangent line is

  • or what the derivative looks like

  • and so let's say when Y is equal to one

  • or when E to the X is equal to one,

  • this is the case when X is equal to zero.

  • Well, the slope of the tangent line

  • looks like it is one, which is curious because

  • that's exactly the value of the function at that point.

  • What about when E to the X is equal to two right over here?

  • Well here, let me do it in another color,

  • the slope of the tangent line sure looks pretty close,

  • sure looks pretty close to two.

  • What about when E to the X is equal to 1/2?

  • So that's happening right about here.

  • Well, it sure looks like the slope of the tangent line

  • is about 1/2.

  • We could try what happens when E to the X

  • is equal to five?

  • Well, the slope of the tangent line here

  • sure does look pretty close,

  • sure does look pretty close to five

  • and so just eyeballing it,

  • is it the case that the slope of the tangent line

  • of E to the X is the same thing, is E to the X?

  • And I will tell you and this is an amazing thing

  • that that is indeed true,

  • that if I have some function, F of X,

  • that is equal to E to the X

  • and if I were to take the derivative of this,

  • this is going to be equal to E to the X as well

  • or another way of saying it,

  • the derivative with respect to X of E to the X

  • is equal to E to the X

  • and that is an amazing thing.

  • In previous lessons or courses,

  • you've learned about ways to define E

  • and this could be a new one.

  • E is the number that where if you take that number

  • to the power of X, if you define a function

  • or expression as E to the X,

  • it's that number where if you take the derivative of that

  • it's still going to be E to the X.

  • And what you're looking here, this curve,

  • it's a curve where the value that's Y value

  • at any point is the same as the slope of the tangent line.

  • If that doesn't strike you as mysterious

  • and magical and amazing just yet, it will.

  • Maybe tonight you'll wake up in the middle of the night

  • and you'll realize just what's going on.

  • Now, some of you might be saying okay, this is cool,

  • you're telling me this, but how do I know it's true?

  • In another video, we will do the proof.

- [Instructor] What we have right over here

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A2 初級

Derivative of __ | Advanced derivatives | AP Calculus AB | Khan Academy

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    yukang920108 に公開 2022 年 07 月 12 日
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