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## Formal definition of the derivative as a limit

# Worked example: Derivative from limit expression

AP.CALC:

CHA‑2 (EU)

, CHA‑2.B (LO)

, CHA‑2.B.2 (EK)

, CHA‑2.B.3 (EK)

, CHA‑2.B.4 (EK)

## Video transcript

The alternate form of the
derivative of the function f, at a number a,
denoted by f prime of a, is given by this stuff. Now this might look a
little strange to you, but if you really think
about what it's saying, it's really just
taking the slope of the tangent line
between a comma f of a. So let's imagine some
arbitrary function like this. Let's say that that
is-- well I'll just write that's our function f. And so you could
have the point when x is equal to a-- this is our
x-axis-- when x is equal to a, this is the point a, f of a. You notice a, f of a. And then we could take the slope
between that and some arbitrary point, let's call that x. So this is the point x, f of x. And notice, the numerator
right here, this is just our change in the
value of our function. Or you could view that as the
change in the vertical axis. So that would give you this
distance right over here. That's what we're doing
up here in the numerator. And then in the
denominator, we're finding the change in
our horizontal values, horizontal coordinates. Let me do that in
a different color. So the change in the horizontal,
that's this right over here. And then they're trying to find
the limit as x approaches a. So as x gets closer and
closer and closer and closer to a, what's going to happen
is, is that when x is out here, we have this secant line. We're finding the slope
of this secant line. But as x gets closer and
closer, the secant lines better and better and
better approximate the slope of the tangent line. Where the limit, as x approaches
a, but doesn't quite equal a, is going to be--
this is actually our definition of
our derivative. Or I guess the alternate form
of the derivative definition. And this would be the slope of
the tangent line, if it exists. So with that all
that out the way, let's try to answer
their question. With the Alternative Form
of the Derivative as an aid, make sense of the
following limit expression by identifying the function
f and the number a. So right here, they want to find
the slope of the tangent line at 5. Here they wanted to find the
slope of the tangent line at a. So it's pretty clear
that a is equal to 5. And that f of a is equal to 125. Now what about f of x? Well here, it's a limit
of f of x minus f of a. Well here it's the limit as
x to the third minus 125. And this makes sense. If f of x is equal
to x to the third, then it makes sense that f of 5
is going to be 5 to the third, is going to be 125. And we're also taking up here
the limit as x approaches a. Here we're taking the
limit as x approaches 5. So this is the derivative
of the function f of x is equal
to x to the third. Let me write that down
in the green color. x to the third at the
number a is equal to 5. And so we can imagine this. Let's try to actually graph it,
just so that we can imagine it. Actually, I'll do
it out here, where I have a little bit better
contrast with the colors. So let's say that is my y-axis. Let's say that
this is my x-axis. I'm not going to quite
draw it to scale. Let's say this right
over here is the 125. Or y, this is when y equals 125. This is when x is equal
to 5, so they're clearly not at the same scale. But the function is going
to look something like this. We know what x to
the third looks like, it looks something like this. So here, our a is equal to 5. This point right
over here is 5, 125. And then we're taking the
slope between that point and an arbitrary x-value. Or I should say an arbitrary
other point on the curve. So this right over here
would be the point, we could call that
x, x to the third. We know that f of x is
equal to x to the third. And let me make it clear. This is a graph of y is
equal to x to the third. And so this expression, right
over here, all of this, this is the slope between
these two points. And as we take the
limit as x approaches 5, so right now this
is our x, as x gets closer and closer
and closer to 5, the secant lines are
going to better and better approximate the slope of the
tangent line at x equals 5. So the slope of a tangent line
would look something like that.