Name: Derivative of __ | Advanced derivatives | AP Calculus AB | Khan Academy
Uploaded: 2022-07-12T12:31:33.000Z
Duration: 3 min 2 s
Description: Thousands of YouTube videos with English-Chinese subtitles! Now you can learn to understand native speakers, expand your vocabulary, and improve your pronunciation...

- [Instructor] What we have right over here

To-infinitive

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

is one of the most fascinating ideas in calculus

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

and think about what the slope of the tangent line is

this is the case when X is equal to zero.

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,

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

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

We could try what happens when E to the X

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

and if I were to take the derivative of this,

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

the derivative with respect to X of E to the X

E is the number that where if you take that number

to the power of X, if you define a function

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

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.

and magical and amazing just yet, it will.

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

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

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

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

expression

fascinating

realize

reinforce