Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007

Jun 27, 2026 03:28 · 49:00 · English · Whisper Turbo · 2 Speakers
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S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
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S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
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S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
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0:19
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So we're ready to begin
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
the fifth lecture.
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
I'm glad to be back.
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Thank you for entertaining my
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
colleague, Haynes Miller.
0:32
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So today we're going to continue where he started.
0:36
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Namely,
0:38
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
what he talked about was the chain rule,
0:41
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
which is probably the most powerful technique for extending the kinds
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
of functions that you can differentiate.
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Now we're going to use the chain rule in some rather clever
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
algebraic ways today.
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So the topic for today is what's known as
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
implicit differentiation.
1:11
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So implicit differentiation is a technique that allows
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
you to differentiate a lot of functions you didn't even know how to find before.
1:20
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And it's a technique,
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
let's wait for a few people to sit down here.
1:26
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Physics,
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
huh?
1:29
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
OK.
1:29
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
More physics.
1:33
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
We'll take a break.
1:37
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
You can get those after class.
1:42
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So we're talking about implicit differentiation.
1:45
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And I'm going to illustrate it by several examples.
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
this is one of the most important and basic formulas that
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
we've already learned part way.
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Namely,
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
the derivative of x to a power is a times
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
x to the a -1.
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Now,
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
what we've done so far,
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
what we've got so far is
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
the exponents 0,
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
plus or minus 1,
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
plus or minus 2.
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Et cetera.
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
You did the integer,
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
positive integer powers in the first lecture.
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And then yesterday,
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Professor Miller told
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
you about the negative powers.
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So what we're going to do right now,
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
today,
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
is we're going to consider the exponents
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
which are rational numbers,
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
ratios of integers.
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So a is m over n.
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
m and n are integers.
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
All right, so that's our goal for right now.
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And we're going to use this method of implicit differentiation.
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
In particular,
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
it's important to realize that this covers the case m equals 1.
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And those are the nth roots.
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So when we take the 1 over n power,
3:06
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
we're going to cover that right now.
3:09
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Along with many other examples.
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So this is our first example.
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So how do we get started?
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Well, we just write down a formula for the function.
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
The function is y is equal to x to the power m over n.
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
That's what we're trying to deal with.
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And now there's really only two steps.
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
take this equation to the nth power.
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So write it as y to the n is equal to x
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
to the m.
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So that's just the same equation,
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
rewritten.
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And now,
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
what we're going to do is we're going to differentiate.
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So we're going to apply d by
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
dx to the equation.
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Now, why is it that we can apply it to the second equation,
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
not the first equation?
4:06
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So maybe I should call these equations 1 and equation 2.
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
The point is that we can apply it to equation 2.
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Now,
4:13
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
the reason is that we don't know how to differentiate
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
x to the m over n.
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
That's something we just don't know yet.
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
But we do know how to differentiate integer powers.
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Those are the things that we took care of before.
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So now we're in shape to be able to do the differentiation.
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So I'm going to write it out explicitly over here without
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
carrying it out just yet.
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
That's d by dx of y to the n is equal to d by dx
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
of x to the m power.
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And now you see
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
this expression here requires us to do something we
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
couldn't do before yesterday.
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Namely,
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
this y is a function of x.
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
rule here.
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So this is the same as,
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
this is by the chain rule now,
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
d by dy of y to the n times
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
dy dx.
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And then on the right -hand side,
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
we can just carry it out.
5:15
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
We know the formula.
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
It's m x to the m minus 1.
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
our scheme.
5:24
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And now you'll see in a minute why we win with
5:28
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
this.
5:29
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So first of all,
5:30
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
there are two factors here.
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
One of them is unknown.
5:33
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
In fact, it's what we're looking for.
5:35
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
But the other one is going to be a known quantity.
5:38
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Because we know how to differentiate y to the n with respect to y.
5:42
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
That's the same formula,
5:43
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
although the letter has been changed.
5:51
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
I'll write it underneath here.
5:52
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
n y to the n -1 dy dx is
5:57
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
equal to m x to the m -1.
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S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
OK,
6:07
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
now comes,
6:09
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
if you like,
6:11
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
the non -calculus part of the problem.
6:15
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Remember, the non -calculus part of the problem is always the messier part of the problem.
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So we want to figure out this formula.
6:22
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
This formula,
6:23
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
the answer over here,
6:24
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
which maybe I'll put in a box now.
6:29
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
It has this expressed much more simply,
6:31
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
only in terms of x.
6:33
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And what we have to do now is to solve for dy dx using algebra,
6:36
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
and then solve all the way in terms of x.
6:39
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So first of all,
6:40
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
we solve for dy dx.
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So I do that by dividing the factor on
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
the left -hand side.
6:47
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So I get here m x to the m -1 divided
6:51
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
by n y to the n -1.
6:54
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And now I'm going to plug in.
7:00
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Right, this is m over n.
7:02
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
This is x to the m minus 1.
7:04
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And now over here I'm going to put in for y x to
7:08
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
the m over n times n minus 1.
7:11
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So
7:15
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
now we're almost done.
7:17
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
But unfortunately we have this mess of exponents that we have to work
7:21
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
out.
7:23
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Write it one more time.
7:24
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So I already recognize the factor a out front.
7:27
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
That's not going to be a problem for me,
7:30
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
and that's what I'm aiming for here.
7:31
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
But now I have to encode all of these powers.
7:34
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So let's just write it out.
7:36
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
It's m minus 1,
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
and then it's minus the quantity n minus 1 times m
7:42
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
over n.
7:43
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So that's the law of exponents
7:48
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
applied to this ratio here.
7:53
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Maybe just over here in the next board.
7:57
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So we have here m minus 1 minus n
8:01
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
minus 1 m over n is equal to
8:05
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
m minus 1.
8:08
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And if I multiply n by this,
8:10
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
I get minus m.
8:11
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And if the second factor is a minus minus,
8:14
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
that's a plus.
8:15
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And that's plus m over n.
8:18
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So altogether,
8:19
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
the two m's cancel.
8:20
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And I have here minus 1 plus m over n.
8:23
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And lo and behold,
8:24
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
that's the same thing as a minus 1.
8:26
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Just what we wanted.
8:27
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So this equals a
8:32
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
x to the a minus 1.
8:33
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Again,
8:37
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
just a bunch of arithmetic.
8:38
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
From this point forward,
8:41
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
from this substitution on,
8:42
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
it's just the arithmetic of exponents.
8:45
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So
8:51
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
we've done our first example here.
8:56
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
I guess I want to give you a couple more examples.
9:00
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So let's just continue.
9:02
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
The next example,
9:04
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
I'll keep it relatively simple.
9:07
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So we have example 2,
9:11
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
which is going to be the function x squared plus y
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S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
squared is equal to 1.
9:18
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Well, that's not really a function.
9:19
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
This,
9:21
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
again, is a way of defining y as
9:25
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
a function of x implicitly.
9:28
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
There's the idea that I could solve for y if I wanted to.
9:32
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And indeed,
9:35
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
let's do that.
9:36
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So if you solve for y here,
9:38
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
what happens is you get y squared is equal to 1 minus
9:42
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
x squared.
9:43
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And y is equal to plus or minus square root of 1
9:47
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
minus x squared.
9:48
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So
9:53
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
this,
9:54
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
if you like,
9:54
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
is the implicit definition.
9:57
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And here is the explicit.
10:00
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
function y is a function of x.
10:02
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And now,
10:04
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
just for my own convenience,
10:06
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
I'm just going to take the positive branch.
10:08
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
This is the function,
10:10
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
let's just draw it right above here.
10:12
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
It's really a circle in disguise.
10:15
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And I'm just going to take the top part of the circle.
10:18
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So we'll take that top hump here.
10:23
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So that means I'm erasing this minus sign.
10:27
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
I'm just taking the positive branch.
10:35
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Just for my convenience,
10:36
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
I could do it just as well with the negative branch.
10:39
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So now I've
10:43
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
taken the solution and I can
10:47
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
differentiate this.
10:48
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So rather than using this dy by dx notation over
10:52
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
here, I'm going to switch notations over here because it's less writing.
10:56
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And I'm going to write y' and change notations.
10:59
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So I want to take the derivative of this.
11:07
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
This is a somewhat complicated function here.
11:10
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
It's the square root of .
11:12
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And the right way always to look at functions like
11:17
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
this is to rewrite them using
11:21
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
the fractional power notation.
11:24
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
That's the first step in computing a
11:29
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
derivative of a square root.
11:30
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And then the second step here
11:34
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
is what?
11:36
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Somebody want to tell me?
11:40
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Chain rule.
11:41
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Chain rule. Yeah, that's good.
11:42
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Chain rule,
11:43
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
that's it.
11:44
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So we have two things.
11:45
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
We start with one and then we do something else to it.
11:47
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So whenever we do two things to something,
11:49
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
we need to apply the chain rule.
11:51
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So 1 minus x squared,
11:52
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
square root.
11:53
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
All right,
11:55
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
so how do we do that?
11:57
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Well, the first factor I claim is the derivative of this thing.
12:01
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So this is a half blah to the minus 1 half.
12:06
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So I'm doing this by the sort of advanced method now because we've already
12:10
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
graduated.
12:11
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
You already did the chain rule last time.
12:13
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So what does this mean?
12:15
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
This is an abbreviation for the derivative
12:20
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
with respect to blah of blah to
12:24
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
the 1 half.
12:25
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Whatever it is.
12:26
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So that's the first factor that we're going to use.
12:30
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Rather than actually write out a variable for it,
12:33
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
and pass through as I did previously with this y and
12:38
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
x variable here,
12:38
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
I'm just going to skip that step and I'm going to let you imagine it
12:42
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
as being a placeholder for that variable.
12:45
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So this variable is now parenthesis.
12:47
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And then I have to multiply that by the rate of change
12:51
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
of what's inside with respect to x,
12:54
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
and that is going to be .
12:56
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
The derivative of is .
13:02
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And now again,
13:04
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
we're glomming everything together here.
13:07
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
We couldn't have done this example two before example one.
13:10
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Because we needed to know that the power rule worked
13:14
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
not just for a integer,
13:17
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
but also for a equals a half.
13:18
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
We're using the case a equals a half right here.
13:22
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
It's one half times,
13:24
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
and this minus a half here is a minus one.
13:29
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Right?
13:29
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So this is the case a equals 1 half.
13:32
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
a minus 1 happens to be minus 1 half.
13:35
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
OK,
13:40
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
so I'm putting all those things together.
13:41
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And you know,
13:42
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
within a week you have to be doing this very automatically.
13:45
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So we're going to do it at this speed now.
13:47
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
You want to do it even faster,
13:49
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
ultimately.
13:49
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Yes?
13:50
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
The question is,
13:54
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
could I have done it implicitly without the square roots?
13:57
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And the answer is yes,
13:59
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
that's what I'm about to do.
14:01
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So this is an illustration of what's called the explicit
14:05
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
solution.
14:07
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So this guy is what's called explicit.
14:10
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And I
14:15
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
want to contrast it with the method that we're going to now use today.
14:18
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So it involves a lot of complications.
14:20
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
It involves the chain rule.
14:21
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And as we'll see,
14:22
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
it can get messier and messier.
14:24
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And then there's the implicit method,
14:26
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
which I claim is easier.
14:28
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So let's see what happens if you do it implicitly.
14:35
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
The implicit method involves,
14:39
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
instead of writing the function in this relatively complicated
14:43
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
way with a square root,
14:44
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
it involves leaving it alone.
14:46
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Don't do anything to it.
14:48
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
In this previous case,
14:51
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
we were left with something which was complicated,
14:53
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
say x to the 1 3rd or x to the 1 1 half or something complicated.
14:57
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
We had to simplify it.
15:00
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Equation 1,
15:00
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
which is more complicated,
15:01
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
we simplified it and then differentiated it.
15:03
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And so that was the simpler case.
15:05
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Well here, the simplest thing for us,
15:07
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
the simplest equation for us to differentiate is the one we started with.
15:11
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Because squares are practically the easiest things after first powers or maybe
15:15
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
0 powers to differentiate.
15:17
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So we're leaving it alone.
15:19
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
This is the simplest possible form for it.
15:21
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And now we're going to differentiate.
15:22
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So what happens?
15:24
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So again,
15:25
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
what's the method?
15:26
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Let me remind you.
15:27
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
You're applying d by dx to the equation.
15:30
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So you have to differentiate the left side of the equation and differentiate
15:34
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
the right side of the equation.
15:35
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So it's this.
15:41
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And what you get is 2x plus 2y
15:47
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
y' is equal to what?
15:49
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
0.
15:51
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Derivative of 1 is 0.
15:53
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So this is the chain rule
15:58
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
again. I did it a different way.
16:00
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
I'm trying to get you used to many different notations at once.
16:03
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Well,
16:04
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
really just 2.
16:05
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Just the prime notation and the dy dx notation.
16:08
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And this is what I get.
16:14
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So now,
16:14
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
all I have to do is solve for y'.
16:18
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So that y',
16:20
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
if I put the 2x on the other side,
16:22
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
is minus 2x and then divide by 2y.
16:26
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Which is minus x over y.
16:28
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So let's compare our solutions.
16:32
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
I apologize,
16:34
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
I'm going to have to erase something to do that.
16:37
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So let's compare our two solutions.
16:44
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
I'm going to put this underneath and simplify it.
16:46
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So what was our solution over here?
16:48
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
It was 1 half 1 minus x squared to
16:52
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
the minus 1 half times minus 2x.
16:56
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
That was what we got over here.
16:58
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And that
17:02
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
is the same thing if I cancel the 2's and I change
17:07
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
it back to looking like a square root.
17:08
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
That's the same thing as minus x divided by square root of 1 minus x squared.
17:16
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
formula for the derivative when I do it the explicit way.
17:19
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And I'll just compare them,
17:22
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
these two expressions here.
17:25
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And
17:29
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
notice that they are the same.
17:31
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
They're the same because y is
17:35
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
equal to square root of 1 minus x squared.
17:38
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Yeah,
17:40
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
question.
17:41
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
The
17:46
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
question is, why did the implicit method not give the bottom half of the circle?
17:50
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Very good question.
17:52
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
The answer to that is that it did.
17:55
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
I just didn't mention it.
17:58
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Wait,
17:59
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
wait, wait, I'll explain.
18:00
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So suppose I'd stuck in a minus sign here.
18:03
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
I would have gotten this with an
18:08
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
extra minus sign.
18:09
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
But then when I compared it to what was over there,
18:12
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
I would have had to have another different minus sign over here.
18:15
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So actually both places would get an extra minus sign.
18:19
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And they would still coincide.
18:20
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So actually the implicit method is a little better.
18:22
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
It doesn't even notice the difference between the branches.
18:24
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
It does the job on both the top and the bottom half.
18:28
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Another way of saying that is that you're calculating the
18:32
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
slopes here.
18:33
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So let's look at this picture.
18:34
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Here's a slope.
18:36
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Let's just take a look at a positive value of x and just check
18:40
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
the signs to see what's happening.
18:42
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
If you take a positive value of x over here,
18:46
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
x is positive,
18:47
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
this denominator is positive,
18:48
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
the slope is negative.
18:48
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
You can see that it's tilting down.
18:50
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So it's OK.
18:53
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Now,
18:54
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
on the bottom side,
18:56
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
it's going to be tilting up.
18:58
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And similarly,
19:00
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
what's happening up here is that both x and y are positive.
19:03
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And this x and this y are positive,
19:05
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
and the slope is negative.
19:07
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
On the other hand,
19:07
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
on the bottom side,
19:08
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
x is still positive,
19:10
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
but y is negative.
19:11
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And it's tilting up because the
19:13
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
The denominator's negative,
19:15
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
the numerator's positive,
19:16
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
and then this minus sign makes it a positive slope.
19:18
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So it matches perfectly in every category.
19:22
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
This is complicated,
19:25
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
however, and it's easier just to keep track of one branch at a time,
19:29
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
even in advanced math.
19:31
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So we only do it one branch at a time.
19:35
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Other questions?
19:43
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
OK, so now I want to give you a slightly more complicated
19:47
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
example here.
19:48
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And indeed,
19:49
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
some of the,
19:50
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
so here's a little more complicated example.
19:54
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
It's not going to be the most complicated example,
19:56
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
but you know,
19:57
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
it'll be a little bit.
20:11
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So this
20:18
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
example,
20:19
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
I'm going to give you a fourth order equation.
20:22
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So y to the fourth plus xy squared minus
20:27
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
2 is equal to 0.
20:28
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Now,
20:32
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
it just so happens that there's a trick to
20:36
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
solving this equation.
20:38
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So actually you can do both the explicit method and
20:42
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
the non -explicit method.
20:45
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
OK, so the explicit method would say,
20:49
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
OK, well I want to solve for this.
20:51
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So I'm going to use the quadratic formula,
20:53
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
but on y squared.
20:55
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
This is quadratic in y squared.
20:57
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Because there's a fourth power and a second power in the first and the
21:01
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
third powers are missing.
21:14
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
times minus 2 divided by 2.
21:18
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So this x is the b,
21:22
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
this minus 2 is the c,
21:24
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
and a is equal to 1 in the quadratic formula.
21:28
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And so the formula for y is
21:32
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
plus or minus the square root of minus
21:36
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
x plus or minus the square root of x squared plus 8
21:40
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
divided by 2.
21:43
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
All right?
21:44
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So now you can see this problem of branches,
21:47
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
this happens actually in a lot of cases,
21:50
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
coming up in an elaborate way.
21:53
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
You have two choices for the sign here,
21:54
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
you have two choices for the sign here.
21:56
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Conceivably there are as many as four roots to this equation because it's a fourth degree
22:00
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
equation.
22:01
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
It's quite a mess,
22:02
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
you should have to check each branch separately.
22:05
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And this really is that level of complexity.
22:09
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And in general,
22:09
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
it's very difficult to figure out the
22:14
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
formulas for quartic equations.
22:17
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
But fortunately,
22:19
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
we're never going to use them.
22:21
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
That is,
22:21
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
we're never going to need those formulas.
22:23
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So the implicit method is
22:28
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
far easier
22:31
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So the implicit method just says,
22:34
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
OK, I'll leave the equation in its simplest form.
22:37
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And now differentiate.
22:40
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So when I differentiate,
22:41
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
I get 4y3 y' plus,
22:46
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
now here I have to apply the product rule.
22:49
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So I differentiate the x and the y squared.
22:55
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
separately.
22:55
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
First I differentiate with respect to x,
22:57
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
so I get y squared.
22:59
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Then I differentiate with respect to the other factor,
23:03
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
the y squared factor.
23:04
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And I get x times 2y y' and
23:08
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
then the 0 gives me 0.
23:10
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So minus 0 equals 0.
23:13
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So there's the implicit
23:17
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
differentiation step.
23:23
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
I just want to solve for y prime.
23:26
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So I'm going to factor out 4y cubed plus
23:30
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
2xy.
23:32
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
That's the factor on y prime.

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