The second example I wanna give
is... and I'm gonna switch notation 
just a little bit so you don't get bored
I'm gonna look at the curve
y equals
1 over x
That's a curve
that uh...
looks like this.
And those curves are called hyperbolas
and you'll notice that x = 0, 
that's not defined.
So, there is no such thing.
Now again, let's write down 
some expectations.
For instance, it's not defined
at 0
because if there's no function, it
can't have a derivative.
Item 2 might be, for instance,
What about for x positive? 
y' is negative.
What about
for x negative?
So, this is a function
whose derivative should always be
negative
.
What should happen
as x goes to infinity?
y' should approach zero,
and similarly,
as x approaches minus infinity
y' should also approach zero,
and there are other things
that should be true, but we won't
do them right now.
So, now let's make the calculation
and see if it fulfills our
expectations.
f(x+h) - f(x) over
h
This is
the magic formula that
we do
over and over and over again
.
So, we have
we have
1 over (x+h)
1 over (x+h)
minus 1 over x
and I don't like these huge fractions.
So, I'm gonna stick the 1 over h
out in front here.
Well, what does this cry out to have done to it?
It says subtract. So, let me subtract.
and put it over a common denominator
and in the numerator here, I get
x - (x+h)
x - (x+h)
the 1 over h comes along 
for the ride
and
this in turn becomes... a little bit 
of
 algebra helps here.
It becomes -h over
(x+h) times x
(x+h) times x
(x+h) times x
and the h's go away
and it becomes -1
-1 over
(x+h) times x
(x+h) times x
And now what do I do?
I let h go to 0
.
Oh, here there's a missing...
I let h go to 0
and this becomes -1 over
x squared.
Correct?
and
So, let me go back. This is the feel
good part
that you find out that indeed
what you though should be true is true
.
Is this thing... it's not defined at 0
,
it's always negative no matter what
,
and is x becomes very big
,
it becomes... goes to zero
,
and as x goes to the other side,
it also
becomes
that way
.
By the way, for my last thing, let's think 
about something we haven't talked about yet,
and that is this.
What else
with -1 over x squared,
what is true about that function? 
Is there something?
Well, if you replace
x by -x, what happens to that?
It stays the same.
So, this thing is the same at x and -x,
which means
that it's symmetrical around the
y-axis
and is that reasonable?
I haven't proved it, but it does seem pretty 
reasonable, doesn't it? That is to say
those two lines have the same slope
or in other words, those two lines are parallel.
and so uh... what we can say is
people use to end their theorems with
QEDs,
