Alternate form of the derivative

So let's think
about how we could

find the slope of
the tangent line

to this curve right
over here, so what

I have drawn in red, at
the point x equals a.

And we've already seen
this with the definition

of the derivative.

We could try to find a
general function that gives us

the slope of the tangent
line at any point.

So let's say we have
some arbitrary point.

Let me define some arbitrary
point x right over here.

Then this would be the
point x comma f of x.

And then we could
take some x plus h.

So let's say that this
right over here is the point

x plus h.

And so this point would be
x plus h, f of x plus h.

We can find the slope
of the secant line that

goes between these two points.

So that would be your
change in your vertical,

which would be f
of x plus h minus f

of x, over the change
in the horizontal, which

would be x plus h minus x.

And these two x's cancel.

So this would be the
slope of this secant line.

And then if we want to find the
slope of the tangent line at x,

we would just take the
limit of this expression

as h approaches 0.

As h approaches 0, this
point moves towards x.

And that slope of the secant
line between these two

is going to approximate the
slope of the tangent line at x.

And so this right over
here, this we would say

is equal to f prime of x.

This is still a function of x.

You give me an arbitrary x
where the derivative is defined.

I'm going to plug it into this,
whatever this ends up being.

It might be some nice,
clean algebraic expression.

Then I'm going to
give you a number.

So for example, if
you wanted to find--

you could calculate
this somehow.

Or you could even
leave it in this form.

And then if you
wanted f prime of a,

you would just substitute a
into your function definition.

And you would say,
well, that's going

to be the limit as h approaches
0 of-- every place you see

an x, replace it with
an a. f of-- I'll

stay in this color for now--
blank plus h minus f of blank,

all of that over h.

And I left those blanks so
I could write the a in red.

Notice, every place where I
had an x before, it's now an a.

So this is the derivative
evaluated at a.

So this is one way to find
the slope of the tangent line

when x equals a.

Another way-- and
this is often used

as the alternate form
of the derivative--

would be to do it directly.

So this is the point
a comma f of a.

Let's just take another
arbitrary point someplace.

So let's say this
is the value x.

This point right over here on
the function would be x comma

f of x.

And so what's the slope of the
secant line between these two

points?

Well, it would be change
in the vertical, which

would be f of x minus f of a,
over change in the horizontal,

over x minus a.

Actually, let me do that
in that purple color.

Over x minus a.

Now, how could we get a better
and better approximation

for the slope of the
tangent line here?

Well, we could take the
limit as x approaches a.

As x gets closer and
closer and closer to a,

the secant line slope is
going to better and better

and better approximate the
slope of the tangent line,

this tangent line that
I have in red here.

So we would want to take the
limit as x approaches a here.

Either way, we're doing
the exact same thing.

We have an expression for
the slope of a secant line.

And then we're bringing those
x values of those points

closer and closer together.

So the slopes of those secant
lines better and better

and better approximate that
slope of the tangent line.

And at the limit, it does become
the slope of the tangent line.

That is the definition
of the derivative.

So this is the more standard
definition of a derivative.

It would give you your
derivative as a function of x.

And then you can then input
your particular value of x.

Or you could use the alternate
form of the derivative.

If you know that,
hey, look, I'm just

looking to find the
derivative exactly at a.

I don't need a
general function of f.

Then you could do this.

But they're doing
the same thing.