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AP.CALC:

CHA‑2 (EU)

, CHA‑2.B (LO)

, CHA‑2.B.2 (EK)

, CHA‑2.B.3 (EK)

, CHA‑2.B.4 (EK)

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. So here, x is equal
to the number e. The point e comma
1 is on the curve. f of e is 1. The natural log of e is 1. And I've drawn the slope
of the tangent line, or I've drawn 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. So the formal
definition wants us to find an expression for the
derivative of our function at any x. So let's say that this is some
arbitrary x right over here. This would be the
point x comma f of x. 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. This right over
here is going to be the point, x plus
h f of x plus h. Now, the whole underlying
idea of the formal definition of limits is to find the
slope of the secant line between these two
points, and then take the limit as
h approaches 0. 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. And the secant line
is going to become a better and better and
better approximation of the tangent line at x. So let's actually do that. So what's the slope
of the secant line? Well, it's the change in
your vertical axis, which is going to be f of x
plus h minus f of x-- over the change in
your horizontal axis. And that's x plus h minus x. And we see here the
difference is just h. Over h. And we're going to
take the limit of that as h approaches 0. So in the case when f of
x is the natural log of x, this will reduce to the
limit as h approaches 0. f of x plus h is
the natural log of x plus h minus the natural log
of x, all of that over h. So this right over here,
for our particular f of x, this is equal to f prime of x. So if we wanted to evaluate
this when x is equal to e, then everywhere we
see an x we just have to replace it with an e. This is essentially
expressing our derivative as a function of x. It's kind of a
crazy-looking function of x. You have a limit
here and all of that. But every place you see an x,
like any function definition, you can replace
it now with an e. So we can-- let me just do that. Whoops. I lost my screen. Here we go. So we could write
f prime of e is equal to the limit as h
approaches 0 of natural log-- let me do it in the
same color so we can keep track of things--
natural log of e plus h-- I'll just leave
that blank for now-- minus the natural log of
e, all of that over h. So just like that. This right over here, if
we evaluate this limit-- if we're able to and
we actually can-- if we are able to
evaluate this limit, this would give us the slope
of the tangent line when x equals e. This is doing the
formal definition. Now let's do the
alternate definition. The alternate
definition-- if you 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 there. So what they say is
hey, look, let's imagine some other x value here. So let's imagine
some other x value. 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. That's x minus e. So that's the slope of the
secant line between those two points. Well, what if you want
to get the tangent line? Well, let's just take the
limit as x approaches e. As x gets closer and
closer and closer, these points are going to get
closer and closer and closer, and the secant line is
going to better approximate the tangent line. So we're just going to take
the limit as x approaches e. So either one of this. This is using the formal
definition of a limit. Let me make it clear that that
h does not belong part of it. So we could either do it
using the formal definition or the alternate definition
of the derivative.

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