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.