U2S2V07 Product rule.txt

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So here we go.
We've got our f and our g measuring
our width and our height of our rectangle.
And f prime and g prime are telling us
how the width and height are changing with respect to time.
And then we're interested in h prime of 3,
which is the rate of change of the area of this rectangle.
And we noticed that there were really these two places.
There was on top and on the right where we're gaining area,
and so we wanted to figure out the rate of change of area
from those two places on the top.
Hopefully you figured out that since we have a 50 meter width
and the top edge is moving up by 2 meters per second,
we just multiply those two things
and we get the rate of change of area on the top.
And on the right, it's a similar sort of thing.
We have 30 meters worth of height.
This right edge is moving out by 4 meters per second.
So if we multiply those, then we get the rate
of increase of area on the right.
And so for h prime overall, what are we going to do?
Well, we'll just add these two things,
and when we do that we get 220.
And notice that the units work out perfectly.
So meters times meters per second
is meters squared per second, and that was our answer.

Now, we're going to want to generalize this.
So let's take a look back and make
sure we know exactly where this 220 number came from.
So in this calculation here, our 50 meters came from f of 3
and the 2 meters per second came from g prime of 3.
The next term, the 30 meters, that was g of 3.
The 4 meters per second, that was f prime of 3.
And so this was the calculation that we
did in order to get our 220.
There was f of 3 times g prime of 3 plus g of 3 times
f prime of 3, and this is what's going
to give us our general rule.
So let me erase this stuff and we'll put it up here.
If h of x is the product of f of x and g of x, then
the derivative of h is going to be equal to f of x times
g prime of x, plus g of x times f
prime of x-- or in other words, the first function times
the derivative of the second function,
plus the second function times the derivative
of the first function.
And this is true at all points where those derivatives exist.
And there we have it-- this is what
we're going to call the product rule.
It's pretty useful.
So let's do an example.
Let's do the example that we started this entire sequence
with.
So h of x equals x to the fourth times sine x.
So here we have a function which is
the product of two functions-- we'll call them
f of x and g of x, and f of x is equal to x to the fourth while
g of x is equal to sine x.
So our product rule tells us how we can differentiate h.
h prime of x we know is going to be
the first function times the derivative
of the second function, plus the second function
times the derivative of the first function.
And we know what the derivatives of f and g are,
so we can just write all these things now.
So we're going to get x to the fourth for f of x.
g prime we know is cosine x, plus g of x we know
is sine x, and then f prime is just 4x cubed.
And there's our answer.
We're done.
So you should take some time now and get some practice using
the product rule on your own.