Derivative as a concept

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

of a slope of a line.

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

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

of a vertical variable

with respect to a horizontal variable,

so for example, here I
have our classic y axis

in the vertical direction and x axis

in the horizontal direction,

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

I could pick two points,

say that point and that point.

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

what is my change in x?"

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

change in x,

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

as change in y over change in x,

so slope is equal to the rate of change

of our vertical variable

over the rate of change of
our horizontal variable,

sometimes described as rise over run,

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

because it has a constant rate of change.

If you took any two points on this line,

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

anywhere they sit on the line,

if you were to do this calculation,

you would get the same slope.

That's what makes it a line,

but what's fascinating

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,

we can think about the rate of change,

the instantaneous rate
of change of a curve,

of something whose rate

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, what would it be?

Well, the average rate of
change between this point and

this point would be the slope of the line

that connects them,

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

but if we picked two different points,

we pick this point and this point,

the average rate of change

between those points all of a
sudden looks quite different.

It looks like it has a higher slope.

So even when we take the
slopes between two points

on the line, the secant lines,

you can see that those
slopes are changing,

but what if we wanted to ask ourselves

an even more interesting question.

What is the instantaneous
rate of change at a point?

So for example, how fast is y changing

with respect to x exactly at that point,

exactly when x is equal to that value.

Let's call it x one.

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 instantaneous rate of change.

So in this case,

the tangent line might
look something like that.

If we know the slope of this,

well then we could say 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,

Usain Bolt example.

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,

and it's known as a derivative,

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

as the instantaneous rate of change.

I'm putting an exclamation mark

because it's so
conceptually important here.

So how can we denote a derivative?

One way is known as Leibniz's notation,

and Leibniz is one of
the fathers of calculus

along with Isaac Newton,

and his notation, you
would denote the slope

of the tangent line

as equaling dy over dx.

Now why do I like this notation?

Because it really comes
from this idea of a slope,

which is change in y over change in x.

As you'll see in future videos,

one way to think about the slope

of the tangent line is, well,

let's calculate the slope of secant lines.

Let's say between that
point and that point,

but then let's get even closer,

say that point and that point,

and then let's get even closer

and that point and that point,

and then let's get even closer,

and let's see what happens as the change

in x approaches zero,

and so using these d's instead of deltas,

this was Leibniz's way of saying,

"Hey, what happens if my changes

in, say, x become close to zero?"

So this idea,

this is known as sometimes
differential notation,

Leibniz's notation, is
instead of just change

in y over change in x,
super small changes in y

for a super small change in x,

especially as the change
in x approaches zero,

and as you will see,

that is how we will
calculate the derivative.

Now, there's other notations.

If this curve is described
as y is equal to f of x.

The slope of the tangent line

at that point could be denoted

as equaling f prime of x one.

So this notation takes a little
bit of time getting used to,

the Lagrange notation.

It's saying f prime is
representing the derivative.

It's telling us the
slope of the tangent line

for a given point,

so if you input an x into
this function into f,

you're getting the corresponding y value.

If you input an x into f prime,

you're getting the slope of
the tangent line at that point.

Now, another notation that
you'll see less likely

in a calculus class but you
might see in a physics class

is the notation y with a dot over it,

so you could write this
is y with a dot over it,

which also denotes the derivative.

You might also see y prime.

This would be more common in a math class.

Now as we march forward
in our calculus adventure,

we will build the tools to
actually calculate these things,

and if you're already
familiar with limits,

they will be very useful,
as you could imagine,

'cause we're really going
to be taking the limit

of our change in y over
change in x as our change

in x approaches zero,

and we're not just going
to be able to figure it out

for a point.

We're going to be able to
figure out general equations

that described the derivative
for any given point,

so be very, very excited.