Lec 4 MIT 18.01 Single Variable Calculus, Fall 2007
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5:00فصل 2: u times v. 125s · Speaker 1
u times v. So the whole new value looks like this. It's u times v . That's the new value. But what's the change in the product? Well, I'd better subtract off what the old value was, which is u times v . OK. Now, according to the rule we're …
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7:05فصل 3: Well, I like that because it's involved the change in u and the change in v. 302s · Speaker 2
Well, I like that because it's involved the change in u and the change in v. So this is equal to delta u times v of x plus delta x minus u times the change in v. Well, I'm almost there. The next step in computing the derivative, take the di…
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12:07فصل 4: v squared. 300s · Speaker 1
v squared. This may be the craziest rule you'll see in this course, but there it is. And I'll try to show you why it's true and see an example. Yeah, there was a hand. Sorry? Uh, ones look the same. u and v look the same. I'll try to make t…
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17:07فصل 5: The reciprocal of a function, 1 over a function. 195s · Speaker 1
The reciprocal of a function, 1 over a function. Here's a copy of my rule. What's du dx in that case? u is a constant. So that term is 0 in this rule. I don't have to worry about this. I get a minus. And then u, u is 1. And dv dx, well, v i…
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20:23فصل 6: How did I do this? 300s · Speaker 2
How did I do this? Right. Where did that x to the minus 2n come from? So the rule, I'm applying this rule. So the denominator in the quotient rule is v squared. And v was x to the n. So the denominator is x to the 2n. And I decided to write…
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25:24فصل 7: But what happens when I let delta t get small? 300s · Speaker 1
But what happens when I let delta t get small? Well, this gives me dy dt. And on the right -hand side I get dy dx times dx dt. So students will often remember this rule, this is the rule, by saying that you can cancel out the dx's. And that…
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30:24فصل 8: it's a composite of two functions. 63s · Speaker 1
it's a composite of two functions. What's the inside function? So I think I'll introduce this new notation. x is 10t. And the outside function is the sine. So y is the sine of x. So now the chain rule says dy dt is Okay, let's see. I take t…
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31:28
You can just think about it in your mind without actually writing it down. d d sin . I'll just do it again without introducing this middle variable explicitly. Think about it. I first of all differentiate the outside function. And I get cos…
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36:28فصل 10: u' also could be denoted du dx. 295s · Speaker 2
u' also could be denoted du dx. I think we've already here today used this. way of rewriting du dx. I think when I was talking about d dt of uv and so on, I pulled that d divided by dt outside and put whatever function you're differentiatin…
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41:32فصل 11: No. So the question is whether the fourth derivative always gives you the original function back, like what happened here. 266s · Speaker 1
No. So the question is whether the fourth derivative always gives you the original function back, like what happened here. No. That's very, very special to sines and cosines. And in fact, let's see an example of that. I'll do a calculation.…