Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007

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S… Speaker 2 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
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S… Speaker 2 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
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S… Speaker 2 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
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S… Speaker 2 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
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S… Speaker 2 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
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0:19
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
I'm Haynes Miller.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
I'm substituting for David Jaroson today.
0:26
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So you have a substitute teacher today.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So I haven't been here in this class with you,
0:44
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
so I'm not completely sure where you are.
0:47
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
I think you've just been talking about differentiation.
0:51
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
And you've got some examples of
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
differentiation,
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
like these basic examples.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
The derivative of x to the n is n x to the n -1.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
And I think maybe you spent some time computing the derivative of
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
the sine function as well recently.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
And I think you have some rules for extending
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
these calculations as well.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
For instance,
1:19
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
I think you know that if you differentiate a
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
constant times a function,
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
what do you get?
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
The
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
constant comes outside like this.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
Or I could write cu' is cu'.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
That's this rule,
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
multiplying by a constant.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
And I think you also know about differentiating a sum.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
Or I could write this as u plus v' is u' plus
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
v'.
2:04
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So I'm going to be using those,
2:08
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
but today I'll talk about a collection of other rules about
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
how to deal with a product of functions,
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
a quotient of functions,
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
and best of all,
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
composition of functions.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
And then at the end I'll have something to say about higher derivatives.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So that's the story for today.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
That's the program.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So let's begin by talking about the product rule.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
the product rule tells you how to differentiate a product of functions.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
And I'll just give you the rule,
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
first of all.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
The rule is it's u 'v plus
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
uv'.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
A little bit funny,
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
differentiating a product gives you a sum.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
But let's see how that works out
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
in a particular example.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
For example,
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
suppose that I wanted to differentiate
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
the product?
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
Well,
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
the product of these two basic examples that we just talked about.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
I'm going to use the same variable in both cases instead of different ones like I did
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
here. So the derivative of x to the n
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
times sin .
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So this is a new thing.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
Couldn't do this without using the product rule.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So the first function is x to the n,
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
and the second one is sin .
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
And we're going to apply this rule.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So u is x to the n,
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
u' is,
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
according to the rule,
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
n .
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
And then I take v and write it down
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
the way it is.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
And then I do it the other way.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
I take u the way it is,
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
that's x to the n,
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
and multiply it by the derivative of v.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
v' we just saw v' is cos .
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So that's it.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
Obviously you can differentiate longer
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
products, products of more things,
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
by doing it one at a time.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
Well, let's see why this is true.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
I want to try to show you why the product's rule holds.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So I think you have a standard way of trying to understand this.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
And it involves looking at the change in
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
the function that you're interested in differentiating.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So I should look at how much the product uv
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
changes when x changes a little bit.
4:43
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
Well,
4:44
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
so how do we compute the change?
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
some new value of x,
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
x plus delta x.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
Well,
4:56
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
I better write down the whole new value of the function.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
u times v.
5:00
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So the whole new value looks like this.
5:03
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
It's u times v .
5:08
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
That's the new value.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
But what's the change in the product?
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S… Speaker 3 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
Well,
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
I'd better subtract off what the old value was,
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
which is u times v .
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
OK.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
Now,
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
according to the rule we're trying to prove,
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
I have to get u' involved.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So I want to involve the change in u alone by itself.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
Let's just try that.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
I see part of the formula for the change in u right there.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
Let's see if we can get the rest of it in place.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So the change in x is u of x plus
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
delta x minus u of x,
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
right?
5:45
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
That's the change in x.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
This part of it occurs up here
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
multiplied by v of x plus delta x.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So let's put that in too.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
Now, this equality sign isn't very good right now.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
I've got this product here so far.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
But I've introduced something I don't like.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
I've introduced u times v ,
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
right?
6:09
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
Minus that.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So the next thing I'm going to do is correct that
6:15
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
little defect by adding in u times v
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
.
6:21
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
Okay,
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
now I've canceled off what was wrong with this line.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
But I'm still not quite there because I haven't put this in yet.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So I better subtract off u
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
times v.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
And then I'll be home.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
But I'm going to do that in a clever way because I notice that I already have
6:42
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
a u here.
6:42
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So I'm going to take this factor of u and
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
make it the same as this factor.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
And so I get u times this minus u times that.
6:52
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
That's the same thing as u times the difference.
6:56
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So that was a little bit strange,
6:59
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
but when you stand back and look at it,
7:00
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
you can see,
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
multiplied out,
7:02
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
the middle terms cancel and you get the right answer.
7:05
S… Speaker 2 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
Well,
7:07
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
I like that because it's involved the change in u and the change in
7:11
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
v. So this is equal to delta u times v
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
of x plus delta x minus u times
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
the change in v.
7:25
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
Well, I'm almost there.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
The next step in computing the derivative,
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
take the difference quotient,
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
divide this by
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
delta
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
x.
8:02
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
If I made a mistake here,
8:04
S… Speaker 2 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
this plus
8:10
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
magically became a minus on the way down here.
8:13
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So I'd better fix that.
8:14
S… Speaker 2 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
Plus
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
u times delta v over delta x.
8:22
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
This u is this u over here.
8:25
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So I've just divided this formula by
8:29
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
delta x.
8:30
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
And now I can take the limit as x goes to 0.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So as delta x goes to 0,
8:37
S… Speaker 2 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
this
8:42
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
becomes the definition of the derivative.
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S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
du dx times,
8:51
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
now,
8:52
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
what happens to this quantity when delta
8:56
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
x goes to 0?
8:57
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So I'm getting closer and closer,
9:01
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
so x is,
9:02
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
so this quantity is getting closer and closer to x.
9:05
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So what happens to the value of v?
9:07
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
It becomes equal to x ,
9:10
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
that uses continuity of v.
9:12
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So v goes to
9:17
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
v by continuity.
9:22
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So this gives me times v,
9:24
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
and then I have u times,
9:26
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
and delta v delta x gives me dv dx.
9:30
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
And that's the formula.
9:31
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
That's the formula as I wrote it down at the beginning over here.
9:34
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
The derivative of a product is given by this
9:38
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
sum.
9:39
S… Speaker 2 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
From
9:50
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
here to here?
9:51
S… Speaker 2 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
Well,
9:52
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
so maybe it's easiest to work backwards and verify that what
9:56
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
I wrote down is correct here.
10:00
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So if you look,
10:01
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
there's a u times v there.
10:04
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
And there's also one here.
10:06
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
And they occur with opposite signs.
10:09
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So they cancel.
10:10
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
And what's left is u times v
10:15
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
minus u
10:19
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
times v.
10:20
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
And that's just what I started with.
10:21
S… Speaker 2 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
I
10:29
S… Speaker 2 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
canceled
10:34
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
out this term and this term.
10:36
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
And what's left is the ends.
10:39
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
Any other questions,
10:40
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
though? Yes.
10:41
S… Speaker 2 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
Well,
10:49
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
I just calculated what delta uv is.
10:54
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
And now I'm going to divide that by delta x on my way to computing
10:58
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
the derivative.
10:58
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
And so I copied down the right -hand side and
11:04
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
divided by delta x.
11:05
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
I just decided to divide the
11:09
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
delta u by delta x and the delta v by delta x.
11:12
S… Speaker 2 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
Good.
11:16
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
Anything else?
11:21
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So we have the product rule here,
11:24
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
the rule for differentiating a product of two functions.
11:26
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
This is making us stronger.
11:28
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
There are many more functions you can find derivatives of now.
11:31
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
How about quotients?
11:32
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
Let's find how to differentiate a quotient of
11:36
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
two functions.
11:36
S… Speaker 2 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
Well,
11:48
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
again, I'll write down what the answer is and then we'll try to verify it.
11:51
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So there's a quotient.
11:53
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
Let me write this down.
11:56
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
There's a quotient of two functions.
11:58
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
And here's the rule for it,
12:00
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
so I always have to think about this and hope that I get it right.
12:03
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
u 'v minus uv' divided by
12:07
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
v squared.
12:08
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
This may be the craziest rule you'll see in this course,
12:11
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
but there it is.
12:12
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
And I'll try to show you why it's true and see an
12:16
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
example.
12:17
S… Speaker 2 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
Yeah,
12:18
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
there was a hand.
12:18
S… Speaker 2 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
Sorry?
12:23
S… Speaker 2 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
Uh,
12:24
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
ones look the same.
12:28
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
u and v look the same.
12:32
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
I'll try to make them look more different.
12:36
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
The v's have points on the bottom.
12:38
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
The u's have little round things on the bottom.
12:40
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
What's the new value of u?
12:42
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
The
12:49
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
value of u at x plus delta x,
12:51
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
it's u plus delta u.
12:54
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
Right?
12:55
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
That's what delta u is.
12:56
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
It's the change in u when x gets replaced
13:00
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
by delta x.
13:01
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
And the change in v,
13:03
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
the new value of v,
13:05
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
is v plus delta v.
13:09
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So this is the new value of u divided by the new value of v.
13:13
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
That's the beginning.
13:15
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
And then I subtract off the old values,
13:18
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
which are u -v.
13:19
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
This will be easier to work this out when I write it out this
13:23
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
way. So now we'll cross multiply,
13:27
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
as I said.
13:27
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So I get uv plus
13:31
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
delta u times v minus,
13:36
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
now I'll cross multiply this way,
13:38
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
you get uv.
13:43
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
minus u times delta v.
13:44
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
And I divide all this by v plus delta v times
13:49
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
u.
13:50
S… Speaker 2 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
Now
13:55
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
the reason I like to do it this way is that you see the cancellation happening here.
13:58
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
uv and uv occur twice,
14:00
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
and so I can cancel them.
14:01
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
And I will.
14:02
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
And I'll answer these questions in a minute.
14:04
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
That's a
14:08
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
v. All right.
14:11
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
Good.
14:14
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
Anything else?
14:15
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
That's what all the hands were.
14:16
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
Good.
14:17
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
All right, so I cancel these.
14:19
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
And what I'm left with then is delta u times v minus
14:23
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
u times delta v.
14:25
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
And all this is over v plus delta v times
14:29
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
v.
14:30
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
OK,
14:31
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
there's the difference.
14:32
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
There's the change in the quotient.
14:35
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
The change in this function is given by this formula.
14:39
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
And now,
14:40
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
to compute the derivative,
14:41
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
I want to divide by delta x and take the limit.
14:44
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So let's write that down.
14:46
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
Delta of uv divided by
14:50
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
delta x is this formula here divided
14:54
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
by delta x.
14:55
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
And again,
14:57
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
I'm going to put the delta x
15:00
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
under these delta u and delta v.
15:02
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
I'm going to put delta x in the denominator,
15:05
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
but I can think of that as dividing into this factor and this factor.
15:09
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So this is delta u over delta x times v
15:14
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
minus u times delta v over
15:19
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
delta x.
15:20
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
And all that is divided by the same denominator,
15:23
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
v plus delta v times v.
15:26
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
Let's put the delta x up in the numerator
15:30
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
there.
15:32
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
Next up,
15:33
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
take the limit.
15:34
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
As delta x goes to 0,
15:36
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
I get,
15:38
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
by definition,
15:39
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
the derivative of u divided by v.
15:42
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
And on the right -hand side,
15:44
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
well, this is the derivative du dx,
15:48
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
right?
15:48
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
times v.
15:51
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
And then u times,
15:54
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
and here it's the derivative dv dx.
15:57
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
Now what about the denominator?
16:04
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So when delta x goes to 0,
16:07
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
v stays the same,
16:09
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
v stays the same,
16:10
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
what happens to this delta v?
16:12
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
It goes to 0 again because v is continuous.
16:16
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So again,
16:19
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
delta v goes to 0 with delta x because
16:23
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
they're continuous and you just get v times v.
16:27
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
I think that's the formula I wrote down over there.
16:30
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
The derivative of u times v minus u times the derivative
16:34
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
of v.
16:35
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
And all divided by the square of the old denominator.
16:38
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
Well,
16:41
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
that's it.
16:41
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
That's the quotient rule.
16:43
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
Weird formula.
16:44
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
Let's see an application.
16:45
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
Let's see an example.
16:47
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So
16:51
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
the example I'm going to give is pretty simple.
16:54
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
I'm going to take the numerator to be just 1.
16:56
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So I'm going to take u equals 1.
17:02
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So now I'm differentiating 1 over
17:07
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
v.
17:07
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
The reciprocal of a function,
17:11
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
1 over a function.
17:11
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
Here's a copy of
17:16
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
my rule.
17:16
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
What's du dx in that case?
17:18
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
u is a constant.
17:22
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So that term is 0 in this rule.
17:25
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
I don't have to worry about this.
17:28
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
I get a minus.
17:29
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
And then u,
17:32
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
u is 1.
17:35
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
And dv dx,
17:36
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
well, v is whatever v is.
17:38
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
I'll write dv dx as v',
17:41
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
and then I get a v2 in
17:45
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
the denominator.
17:45
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So that's the rule.
17:47
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
I could write it as
17:51
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
v to the minus 2 v'.
17:53
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
minus v'
17:57
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
divided by v2.
17:58
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
That's the derivative of 1 over v.
18:00
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
How about even a
18:05
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
sub -example of that?
18:11
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
I'm going to take the special case where u is equal to 1 again and
18:16
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
v is a power of x.
18:19
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
And I'm going to use the rule that
18:23
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
we developed earlier about the derivative of x to the n.
18:27
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So what do I
18:31
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
get here?
18:31
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
d over dx of 1 over
18:35
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
x to the n is,
18:39
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
I'm plugging into this formula here,
18:41
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
with v equals x to the n.
18:44
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So I get minus v
18:49
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
to the minus 2.
18:50
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
If v is x to the n,
18:53
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
v to the minus 2 is,
18:54
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
by the rule of exponents,
18:55
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
x to the minus 2 n.
18:58
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
And then v' is
19:02
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
the derivative of x to the n,
19:04
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
which is n x to the n minus 1.
19:10
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So let's put these together.
19:11
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
There's several powers of x here.
19:13
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
I can put them together.
19:14
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
I get n times x
19:18
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
to the .
19:19
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
One of these n's cancels,
19:23
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
and what I'm left with is .
19:25
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So we've
19:30
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
computed the derivative of 1 over x to the n,
19:32
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
which I could also write as x to the minus n.
19:38
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So I've computed the derivative of negative powers of x,
19:42
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
and this is the formula that I get.
19:44
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
If you think of this minus n as
19:48
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
a unit,
19:49
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
as a thing to itself,
19:51
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
it occurs here in the exponent,
19:53
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
it occurs here,
19:55
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
and it occurs here.
20:00
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So how does that compare with the formula that we had up here?
20:02
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
The derivative of a power of x is that
20:07
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
power times x to one less than that power.
20:10
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
That's exactly the same as the rule that I wrote down here.
20:14
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
But the power here happens to be a negative number.
20:18
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
And the same negative number shows up as a coefficient.
20:21
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
And there in the exponent.
20:23
S… Speaker 2 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
How
20:30
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
did I do this?
20:47
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
Right.
20:47
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
Where did that x to the minus 2n come from?
20:51
S… Speaker 2 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So
20:56
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
the rule, I'm applying this rule.
20:58
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So the denominator in the quotient rule is v
21:02
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
squared. And v was x to the n.
21:05
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So the denominator is x to the
21:09
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
2n.
21:10
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
And I decided to write it as x to the minus 2n.
21:18
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So the green comments there,
21:21
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
what they say is that I can enlarge this rule.
21:26
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
This exact same rule happens is true for negative
21:30
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
values of n as well as positive values of n.
21:33
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So there's something new
21:37
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
in your list of rules
21:42
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
that you can apply of values of the derivative.
21:44
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
That standard rule is true for negative as well as
21:48
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
positive exponents.
21:49
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
And that comes out of the quotient rule.
21:57
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
OK, so we've done two rules.
21:58
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
I've talked about the product rule and the quotient rule.
22:03
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
What's next?
22:05
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
Let's do the chain rule.
22:06
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So this is a composition rule.
22:08
S… Speaker 2 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So
22:22
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
the kind of thing that I have in mind,
22:24
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
composition of functions is about substitution.
22:27
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So the kind of function that I have in mind is,
22:30
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
for instance, y is the sine of
22:34
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
t raised to the tenth power.
22:37
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
That's a new one.
22:42
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
I haven't seen how to differentiate that before,
22:44
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
I think.
22:45
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
This kind of power of a trig function
22:49
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
happens very often.
22:50
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
You've seen them happen as well,
22:52
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
I'm sure, already.
22:53
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
And there's a little notational switch that people use.
22:57
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
They'll write sin to the 10 of t.
23:01
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
But remember that when you write sin to the 10 of t,
23:05
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
what you mean is take the sin of t and then take the tenth power
23:09
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
of that.
23:09
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
That's the meaning of sin to the 10 of t.
23:13
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So the method of
23:18
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
dealing with this kind of composition of functions is
23:22
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
to introduce u's new variable
23:26
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
names.
23:27
S… Speaker 2 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
What
23:33
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
I mean is,
23:34
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
I can think of this sin raised
23:38
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
to the tenth power,
23:39
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
I can think of it as a two -step process.
23:41
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
First of all,
23:42
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
I compute the sin and let's call the result x.
23:45
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
There's the new variable name.
23:49
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
And then I express y in terms of x.
23:53
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So y says take this and raise it to the 10th
23:57
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
power.
23:57
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
In other words,
23:58
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
y is the function x to the 10th power.
24:01
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
And then you plug x equals sin into
24:05
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
that and you get the formula for what y is in terms of t.
24:09
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So it's good practice to introduce
24:13
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
new letters when they're convenient.
24:15
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
And this is one place where it's very convenient.
24:21
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So let's find a rule for differentiating a composition,
24:24
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
a function that can be expressed by doing one function and then applying another function.
24:29
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
And here's the rule.
24:31
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
Well,
24:33
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
maybe I'll actually derive this rule first and then you'll see what it is.
24:36
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
In fact,
24:37
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
the rule is very simple to derive.
24:40
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
So this is a proof first and then we'll write down the rule.
24:43
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
I'm interested in delta y over
24:47
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
delta t.
24:50
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
y is a function of x,
24:51
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
x is a function of t.
24:53
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
And I'm interested in how y changes with respect to t,
24:57
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
with respect to the original variable t.
25:00
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
Well, because of that intermediate variable
25:05
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
I can write this as delta y divided by delta x
25:09
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
times delta x divided by delta t.
25:11
S… Speaker 2 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
It cancels,
25:13
S… Speaker 1 (Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007)
right? The delta x cancels.

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