okay in this video I want to talk about just
making sense out of this definition of the derivative.
So, the definition of the derivative is down here at the bottom.
We say the derivative f'(x) [f prime of x]. So this is our notation for the derivative.
The definition of the derivative
is, we say it's the limit as h goes to 0 of f(x+h) minus f(x)
divided by h, and I'm going to try to make a tie between this definition
as related to slopes of tangent lines. I think a lot of people just remember this,
you know memorize it and aren't really sure where it comes from so again
somehow this definition represents the slope
of a tangent line at a particular X coordinate
ok, so somehow that's what all this represents so
it's going to give you the slope
of the tangent line at
some X coordinate. okay so let's try to make some sense out of this so
I've got a little picture here forgive my artistry
so again in black I've got my function y = f(x)
and (let me make that a little prettier)
so y=f(x) and the idea is, ok, if you're going along this curve our
tangent line would be the line here in red
the way I always thought about the tangent line was you know if I was on a
roller coaster
(you know, so there's my little roller coaster) I wish it was a little shorter
if I was sitting on that point basically I think whatever my line of
sight would be that would be
what the tangent line would look like so for example at the very tip top
when you're at the highest point on the roller coaster
if you're looking straight ahead your line of sight you know the tangent line
would just be a horizontal line
and then as you start moving down the roller coaster
well the tangent line would be pointing downwards. Here at this point in red
that I have marked it looks like the graph is going a little bit upwards
so my line of sight would be something like as indicated by the red
tangent line. okay so again what we're interested in here
is we want to find a formula again for the slope
of this tangent line
and the way we do it is through a roundabout way
that makes use of limits and the idea
what we're going to do is instead of calculating slopes at the tangent line
because again to calculate the slope of the line
we need to know two points that the line goes through. Well
I'm making it you know I'm picking an arbitrary point again let's call that
X, so I've got one point that the line goes through but I'm not really sure
you know about other points but that's okay what we do is
we take ... what we're going to do is we're going to find
the slope between two points on the graph
so I've got marked in blue (they kind of look a little black) I've got three points marked here in
blue and what I'm going to do is instead of
you know finding the slope of the tangent line I'm going to find the slope of the
line
between my original point and one of these new points
when you take two points on the graph and you connect to them you find that line
that's what's known as a secant line
Ok so I've got one little secant line here
okay and you know I could calculate the slope of this line because now I know
two points that the line goes through
well ok who cares the idea is what we're gonna do you instead is
instead of going - again I'm interested in the tangent line at this point -
well we found a secant line. this new point is kind of far away from my original
point
what I'm going to start doing is I'm going to start bringing the point
closer to the place where I'm trying to find
the tangent line okay so we could calculate
the slope of yet another secant line
and then we could calculate yet the slope of
another secant line okay so my graph is going to get a little cluttered here
pretty quickly
the idea intuitively is, you know, if you look at my first line:
okay it doesn't look like the tangent line very much at all but as I start
bringing this point in closer
and closer and closer this
secant line is going to start to resemble
the tangent line. okay so
again this is the key idea: if you bring the point
in closer to the point where you are trying to find the tangent line
the idea is that those secant lines - the blue lines - are going to get closer and
closer and closer and closer
to where they look almost exactly like the tangent line
so really what we're doing is okay so we're going to find
slopes of secant lines
okay and what we're gonna do is - really what we're doing so maybe I say
that this first secant line
- the first one I draw up there - suppose that has slope s1
and the second line has slope s2
and the third line has slope s3 and what I'm going to do is I'm going to keep
making point - I'm going to keep picking points closer and closer and closer and
closer
the way that we define the slope of the tangent line
we say the slope of the tangent line
is going to equal the limit
of the slopes
of the secant lines. okay so this is the key idea
okay so we're going to make the slope of the tangent line equal to the limit of
the slopes of the secant lines
well okay so now we need to somehow -so this is the first thing you want to
make sure that
hopefully this idea hopefully I have explained it fairly clearly
bring the point in closer closer closer - the secant lines are going to start to
resemble
the tangent line
so this is kind of the first key idea
from here we simply need to take this and kind of make it
mathematical. okay, so I'm going to erase this picture
and try to redraw something similar
okay so keep this idea in mind we're going to revisit that
okay so here's my graph hopefully
it will sort of resemble what I just had getting pretty close
okay so again we're interested in finding
were interested in finding the slope of this tangent line
but to do that we're going to calculate the slopes of secant lines
okay so here's where we're going to introduce some notation to try to make some sense
out of this
Ok, so I'm going to connect these two points again just so I have at least one secant
line in there
okay so we're trying to find the slope of the tangent line let's just call this
point generically
some X coordinate - well
the notation that we're going to use is -
so I've got to know the coordinates of this other point
let's just suppose it's some distance away and it is certainly some distance away
from
our X coordinate - let's call that
distance that it's away - let's call that the value
h - so again h just simply represents a distance
from the point where you want to find the slope of the tangent
line to - it's the distance to this new point where we're going to calculate the
slope
of the secant line - well
if we're originally sitting at the point X
this new point is going to be, well, X
plus h, because that's how far we've moved away and now we can label these
points generically on our graph
so we know we're at the x-coordinate X
the y coordinate would be whatever f(x) is -
well likewise if we think about
this other point, this new point
it would have an x coordinate of, well, x+h
equivalently its y coordinate would be whatever f(x+h)
is equal to. so what that means is it says the slope of the secant line
it says the slope of the secant line - well that's just change in Y minus change
in X - so I'm going to take my y coordinate f(x+h)
so there's f(x+h) minus the other y coordinate which would be
f(x)
and then we have to do the change in X so if I take my x coordinate which is now
x+h and I subtract away the original X coordinate which is x
I've now got a generic formula for the slope of the secant line
notice you can get rid of the parentheses in the bottom you have X minus
X and you would just be left simply with H
so hey I'm getting pretty close here now so there's our
f(x+h) minus f(x) again all this part really represents
is the slope of the secant line that's all this
f(x+h) minus f(x) over h is
it's the slope of a secant line
okay so we're going to define the slope
of the tangent line by using slopes
of secant lines so the notation for the slope of the tangent line at any point we use
this little f'(x) [f prime of x]
okay these aren't quite equal yet so I have to be careful
but if this is the slope of the secant line - what would we want to happen?
we wanted to take limits of these slopes
well we wanted something else to happen as well we wanted the points
on the line to get closer and closer
and closer and closer to our original X coordinate
so we said we're taking the limit
but to make the points get closer and closer and closer and closer and closer
and closer and closer
what would have to happen to our H value well
our h value would have to get smaller
and smaller and smaller if we're going to bring it in closer
and closer and closer
to get the corresponding
values on the secant line. if I bring this point in closer and closer and
closer
well what's going to happen H well h is going to get smaller and smaller and smaller
and it's going to approach this value: Zero
so "Tada!" we now have produced a formula
that would give us - the slope of the tangent line
at any point - again all we're recognizing is
were taking the limit of slopes of secant lines
and if you can remember that this thing on the right really represents just the slope
of a secant line
we're trying to bring the points in closer that's why h approaches 0
well, now you have it memorized, now you understand why we use this particular
definition it looks awkward at first but
hopefully after going through this its it makes you think it's exactly what it
should be
alright again I think this idea can be a little confusing
so if you don't catch everything at once you know that's normal. feel free to post
comments or questions hopefully me or somebody else can point you in the right
direction