U2S2V10 The Product Rule, Formally.txt

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In this video, we're going to derive the product
rule just a bit more formally.
We've got h of t as the product f of t times g of t,
and for h prime of t, we're going to use delta notation.
So remember that h prime of t is approximately
equal to delta h over delta t.
And then to make this exact, we're
ultimately going to want to take the limit as delta t goes to 0.
So let's look at this numerator.
What is delta h?
Well, it's the change in h, so there's a new h
and we subtract the old h.
But what does that mean?
Well, h has changed because f and g have changed.
h is always f times g, so the new h should
be the new f-- that's our original f--
plus the change in f, delta f, times the new g,
so that's the original g plus the change in g.
This product, then, is the new h.
We subtract the old h, which is just the original f times g.
We can visualize all of that with the rectangle picture.
h equals f times g like this, and our new h
is the area of a new rectangle.
The width changes by delta f and the height changes by delta g.
And then the new h is the area of this new rectangle,
and that's f plus delta f times g plus delta g.
And then in this picture, delta h, the change in area,
that's everything in this region right here.
But notice one thing.
In this picture, we're assuming that all of these quantities
are positive, so f, delta f, et cetera.
With the algebra, there's no such assumption,
so the algebra is more general and more formal.
The picture is just here to help us get an intuition
about all of these terms.

Back to the algebra, then, if we multiply this out,
we're going to get f times g plus f times delta g
plus g times delta f plus delta f times delta g,
and then we subtract f times g.
So these things cancel and we're left
with f times delta g plus g times delta f plus delta f
times delta g.
Now, this is supposed to be delta h.
And in fact, we can see in the picture
where all three of these terms come from.
f times delta g, that's this area right here.
g times delta f is this part of the delta h, and delta
f times delta g, that's this section up in the corner.
If you are worried that we missed this corner
piece in the last video about the product rule, don't worry.
The algebra sees it and the algebra will tell us
that it really isn't a problem.

We've got our delta h and we're supposed to divide that
by delta t, and I'm going to write this
in a form that's suggestive.
When we divide this first term by delta t,
we're going to get f times delta g over delta t.
Then for the next term, we'll put g times delta f
over delta t.
And the last term, I'm going to do something funny here.
We've got delta f over delta t, that's dividing by delta t,
and then I'm going to put delta g over delta t again,
and then that's an extra delta t in the denominator
so I have to multiply by delta t.
That's how I'm going to write that.
Now remember, we're supposed to take the limit as delta t goes
to 0, and we're going to do that to all of these terms.
And in the first one, well we've got f.
That stays the same, but what's going on
with delta g over delta t as delta t goes to 0?
Well, that's just the definition of the derivative.
That's g prime.
We're assuming that g is differentiable here.
And in the next term, we'll get g times-- and then
this quotient.
Well, that approaches f prime as delta t approaches 0.
And finally, in the last term, the first part,
that's going to go to f prime.
The second part goes to g prime, but the last part, well, that's
just delta t itself, which is going to 0.
So in the limit, this last term just disappears-- it's 0--
and we're left with f times g prime plus g times f prime.
Hey.
That's our product rule.