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18.01 Single Variable Calculus, Fall 2007
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18.01 Single Variable Calculus, Fall 2007
Transcript – Lecture 4
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Professor: I am Haynes Miller, I am substituting for David Jerison today. So you
have a substitute teacher today. So I haven't been here in this class with you so I'm
not completely sure where you are. I think just been talking about differentiation and
you've got some examples of differentiation like these basic examples: the derivative
of x^n is nx^(x1). But I think maybe you've spent some time computing the
derivative of the sine function as well, recently. And I think you have some rules for
extending these calculations as well. For instance, I think you know that if you
differentiate a constant times a function, what do you get?
Student: [INAUDIBLE].
Professor: The constant comes outside like this. Or I could write (cu)' = cu'. That's
this rule, multiplying by a constant, and I think you also know about differentiating a
sum. Or I could write this as (u v)' = u' v'. So I'm going to be using those but
today I'll talk about a collection of other rules about how to deal with a product of
functions, a quotient of functions, and, best of all, composition of functions. And then
at the end, I'll have something to say about higher derivatives. So that's the story
for today. That's the program.
So let's begin by talking about the product rule. So the product rule tells you how to
differentiate a product of functions, and I'll just give you the rule, first of all. The rule
is it's u'v
uv'. It's a little bit funny. Differentiating a product gives you a sum. But
let's see how that works out in a particular example. For example, suppose that I
wanted to differentiate the product. Well, the product of these two basic examples
that we just talked about. I'm going to use the same variable in both cases instead
of different ones like I did here. So the derivative of (x^n)sin x. So this is a new
thing. We couldn't do this without using the product rule. So the first function is x^n
and the second one is sin x. And we're going to apply this rule. So u is x^n. u' is,
according to the rule, nx^(n  1). And then I take v and write it down the way it is,
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This note was uploaded on 01/18/2012 for the course MATH 18.01 taught by Professor Brubaker during the Fall '08 term at MIT.
 Fall '08
 BRUBAKER
 Calculus

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