Name: Derivative as a concept | Derivatives introduction | AP Calculus AB | Khan Academy
Uploaded: 2022-07-12T11:58:31.000Z
Duration: 7 min 16 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...

Adverbs of manner

- [Instructor] You are likely already familiar with the idea

If you're not, I encourage you to review it on Khan Academy,

but all it is, it's describing the rate of change

so for example, here I have our classic y axis

and if I wanted to figure out the slope of this line,

I could say, "Okay, from this point to this point,

Well, my change in x would be this distance right over here,

the Greek letter delta, this triangle here.

It's just shorthand for "change," so change in x,

and I could also calculate the change in y,

so this point going up to that point, our change in y,

would be this, right over here, our change in y,

and then, we would define slope, or we have defined slope

over the rate of change of our horizontal variable,

and for any line, it's associated with a slope

because it has a constant rate of change.

no matter how far apart or no matter how close together,

about calculus is we're going to build the tools

so that we can think about the rate of change not just

of a line, which we've called "slope" in the past,

the instantaneous rate of change of a curve,

of change is possibly constantly changing.

So for example, here's a curve where the rate of change of y

with respect to x is constantly changing,

even if we wanted to use our traditional tools.

If we said, "Okay, we can calculate the average rate

of change," let's say between this point and this point.

Well, the average rate of change between this point and

this point would be the slope of the line

so it would be the slope of this line of the secant line,

between those points all of a sudden looks quite different.

So even when we take the slopes between two points

you can see that those slopes are changing,

What is the instantaneous rate of change at a point?

Well, one way you could think about it is

what if we could draw a tangent line to this point,

a line that just touches the graph right over there,

and we can calculate the slope of that line?

Well, that should be the rate of change at that point,

the tangent line might look something like that.

that's the instantaneous rate of change at that point.

Why do I say instantaneous rate of change?

Well, think about the video on these sprinters,

If we wanted to figure out the speed of Usain Bolt

at a given instant, well maybe this describes his position

with respect to time if y was position and x is time.

Usually, you would see t as time, but let's say x is time,

so then, if were talking about right at this time,

we're talking about the instantaneous rate,

and this idea is the central idea of differential calculus,

the slope of the tangent line, which you could also view

because it's so conceptually important here.

and Leibniz is one of the fathers of calculus

and his notation, you would denote the slope