Name: Formal and alternate form of the derivative for ln x | Differential Calculus | Khan Academy
Uploaded: 2022-07-12T12:05:40.000Z
Duration: 5 min 45 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...

Let's say that f of x is equal to the natural log of x,

and we want to figure out what the slope of the tangent line

to the curve f is when x is equal to the number e.

And I've drawn the slope of the tangent line,

And we need to figure out what the slope of it

is, or at least come up with an expression for it.

And I'm going to come up with an expression using

both the formal definition and the alternate definition.

That will allow us to compare them a little bit.

So let's think about first the formal definition.

to find an expression for the derivative of our function

So let's say that this is some arbitrary x right over here.

And let's say that this is-- let's call this x plus h.

So this distance right over here is going to be h.

Now, the whole underlying idea of the formal definition

of limits is to find the slope of the secant line

As h gets closer and closer, this blue point

is going to get closer and closer and closer to x.

And this point is going to approach it on the curve.

a better and better and better approximation

Well, it's the change in your vertical axis, which

is going to be f of x plus h minus f of x--

And we see here the difference is just h.

And we're going to take the limit of that

So in the case when f of x is the natural log of x,

this will reduce to the limit as h approaches 0.

plus h minus the natural log of x, all of that over h.

So this right over here, for our particular f of x,

So if we wanted to evaluate this when x is equal to e,

This is essentially expressing our derivative

It's kind of a crazy-looking function of x.

But every place you see an x, like any function definition,

equal to the limit as h approaches 0 of natural log--

can keep track of things-- natural log of e plus h--

minus the natural log of e, all of that over h.

This right over here, if we evaluate this limit--

this would give us the slope of the tangent line when

don't want to find a general derivative expressed

as a function of x like this and you just

want to find the slope at a particular point,

the alternate definition kind of just gets straight to the point

So what they say is hey, look, let's imagine

This right over here is the point x comma-- well,

we could say f of x or we could even say the natural log of x.

What is the slope of the secant line between those two points?

Well, it's going to be your change in y values.

So it's going to be natural log of x minus 1--

let me do that red color-- over your change in x values.

So that's the slope of the secant line between those two

Well, what if you want to get the tangent line?

Well, let's just take the limit as x approaches e.

these points are going to get closer and closer and closer,

and the secant line is going to better approximate