Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007
Дыбыс тақшалары
Бөлімдер
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5:00Бөлім 2: rule here. 60s · Speaker 1
rule here. So this is the same as, this is by the chain rule now, d by dy of y to the n times dy dx. And then on the right -hand side, we can just carry it out. We know the formula. It's m x to the m minus 1. our scheme. And now you'll see …
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OK, now comes, if you like, the non -calculus part of the problem. Remember, the non -calculus part of the problem is always the messier part of the problem. So we want to figure out this formula. This formula, the answer over here, which m…
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11:07Бөлім 4: This is a somewhat complicated function here. 301s · Speaker 1
This is a somewhat complicated function here. It's the square root of . And the right way always to look at functions like this is to rewrite them using the fractional power notation. That's the first step in computing a derivative of a squ…
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16:08Бөлім 5: And this is what I get. 294s · Speaker 1
And this is what I get. So now, all I have to do is solve for y'. So that y', if I put the 2x on the other side, is minus 2x and then divide by 2y. Which is minus x over y. So let's compare our solutions. I apologize, I'm going to have to e…
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21:14Бөлім 6: times minus 2 divided by 2. 305s · Speaker 2
times minus 2 divided by 2. So this x is the b, this minus 2 is the c, and a is equal to 1 in the quadratic formula. And so the formula for y is plus or minus the square root of minus x plus or minus the square root of x squared plus 8 divi…
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26:20Бөлім 7: this formula star here to find y. 98s · Speaker 1
this formula star here to find y. Now, so let me just make two points about this, which are just philosophical points for you right now. The first is, when I promised you at the beginning of this class that we were going to be able to diffe…
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27:58Бөлім 8: Yes? What's in the parentheses? 298s · Speaker 2
Yes? What's in the parentheses? Sorry. Let's see if I can manage. Is this the parentheses you're talking about? Well, so maybe I should put commas around it. But it was s, a, y, comma, comma. Well, here was at x equals 1. And here's but at,…
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axes, then that would be, in effect, trading the roles of x and y. We have to rename every point on the graph, which is the ordered pair x, y, and trade it for the opposite ones. And when you exchange x and y, so to do this, exchange x and …
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38:00Бөлім 10: function tan x. 300s · Speaker 1
function tan x. And so the one that we have to sketch is this one which we get by reflecting this. across the axis. So that's going to be, well, not the axis, sorry, the diagonal. This slope, by the way, should be less, a little lower here …
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43:01Бөлім 11: of tan inverse of x. 301s · Speaker 1
of tan inverse of x. Now, so far we have the following answer to that question. It's cosine squared of tan inverse x. Now, this is a correct answer, but way too complicated. Now, that doesn't mean that if you took a random collection of fun…
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48:02Бөлім 12: I have to just recognize that this is the same as this, which is the same as 1 over square root of . 54s · Speaker 1
I have to just recognize that this is the same as this, which is the same as 1 over square root of . So all told, the derivative with respect to x of the arc sine function is 1 over square root of . So these implicit differentiations are ve…