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18-01F07-L04 - MIT OpenCourseWare http/ocw.mit.edu 18.01...

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MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus, Fall 2007 Please use the following citation format: David Jerison, 18.01 Single Variable Calculus, Fall 2007 . (Massachusetts Institute of Technology: MIT OpenCourseWare). http://ocw.mit.edu (accessed MM DD, YYYY). License: Creative Commons Attribution-Noncommercial-Share Alike. Note: Please use the actual date you accessed this material in your citation. For more information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms
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MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus, Fall 2007 Transcript – Lecture 4 The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation, or to view additional materials from hundreds of MIT courses visit MIT OpenCourseWare at ocw.mit.edu. 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^(x-1). 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, sine of x. And then I do it the other way. I take u the way it is, that's x^n, and multiply it by the derivative of v, v'. We just saw v' is cosine of x. So that's it.
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