about what the derivative with respect to x

of the natural log of x's.

And I'm gonna go straight to the punch line.

It is equal to one over x.

In a future video, I'm actually going to prove this.

It's a little bit involved.

But in this one, we're just going to appreciate

that this seems like it is actually true.

So right here is the graph

of y is equal to the natural log of x.

And just to feel good about the statement,

let's try to approximate

what the slope of the tangent line is at different points.

So let's say right over here,

when x is equal to one,

what does the slope of the tangent line look like?

Well, it looks like here,

the slope looks like it is equal,

pretty close to being equal to one,

which is consistent with the statement.

If x is equal to one, one over one is still one,

and that seems like what we see right over there.

What about when x is equal to two?

Well, this point right over here is the natural log of two,

but more interestingly, what's the slope here?

Well, it looks like,

let's see, if I try to draw a tangent line,

the slop of the tangent line looks pretty close to 1/2.

Well, once again, that is one over x.

One over two is 1/2.

Let's keep doing this.

If I go right over here, when x is equal to four,

this point is four comma natural log of four,

but the slope of the tangent line here

looks pretty close to 1/4

and if you accept this, it is exactly 1/4,

and you could even go to values less than one.

Right over here, when x is equal to 1/2,

one over 1/2, the slope should be two.

And it does indeed, let me do this

in a slightly different color,

it does indeed look like the slope is two over there.

So once again, you take the derivative

with respect to x of the natural log of x, it is one over x.

And hopefully, you get a sense

that that is actually true here.

In a future video, we'll actually prove it.