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PROFESSOR: OK.
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This video is about
derivatives.
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Two rules for finding
new derivatives.
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If we know the derivative
of a function f--
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say we've found that--
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and we know the derivative
of g--
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we've found that--
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then there are functions that
we can build out of those.
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And two important and
straightforward ones are the
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product, f of x times g of x,
and the quotient, the ratio f
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of x over g of x.
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So those are the two
rules we need.
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If we know df dx and we know dg
dx, what's the derivative
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of the product?
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Well, it is not df
dx times dg dx.
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And let me reduce the suspense
by writing down what it is.
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It's the first one times the
derivative of the second, we
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know that, plus another term,
the second one times the
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derivative of the first. OK.
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So that's the rule to learn.
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Two terms, you see
the pattern.
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And maybe I ought to use it,
give you some examples, see
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what it's good for, and
also some idea of
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where it comes from.
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And then go on to the
quotient rule,
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which is a little messier.
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OK.
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So let me just start by using
this in some examples.
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Right underneath, here.
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OK.
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So let me take, as a first
example, f of x equals x
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squared and g of x equals x.
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So then what is p of x?
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It's x squared times x.
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I'm multiplying the functions.
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So I've got x cubed, and I want
to know its derivative.
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And I know the derivatives
of these guys.
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OK, so what does the
rule tell me?
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It tells me that the derivative
of p, dp dx--
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so p is x cubed.
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So I'm looking for the
derivative of x cubed.
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And if you know that, it's OK.
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Let's just see it
come out here.
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So the derivative of x cubed,
by my formula there, is the
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first one, x squared, times the
derivative of the second,
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which is 1, plus the second one,
x, times the derivative
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of the first, which is 2x.
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So what do we get?
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x squared, two more x
squared, 3x squared.
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The derivative of x cubed
is 3x squared.
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x cubed goes up faster than
x squared, and this
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is a steeper slope.
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Oh, let's do x to the fourth.
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So x to the fourth-- now I'll
take f to be x cubed, times x.
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Because x cubed, I just found.
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x, its derivative is 1, so I
can do the derivative of x
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fourth the same way.
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It'll be f.
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So practicing that formula again
with x cubed and x, it's
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x cubed times 1 plus this guy
times the derivative of f.
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Right?
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I'm always going back
to that formula.
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So the derivative of f, x
cubed, we just found--
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3x squared-- so I'll
put it in.
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And what do we have?
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x cubed here, three more
x cubeds here.
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That's a total of 4x cubed.
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OK.
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We got another one.
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Big deal.
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What is important is--
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and it's really what
math is about--
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is the pattern, which we can
probably guess from those two
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examples and the one we
already knew, that the
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derivative of x squared
was 2x.
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So everybody sees a 2 here and
a 3 here and a 4 here, coming
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from 2, 3, and 4 there.
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And everybody also sees that
the power dropped by one.
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The derivative of x
squared was an x.
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The derivative of x cubed
involved an x squared.
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Well, let's express this
pattern in algebra.
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It's looking like the derivative
of x to the n--
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we hope for any n.
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We've got it for n equals 2,
3, 4, probably 0 and 1.
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And if the pattern continues,
what do we think?
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This 4, this n shows up there,
and the power drops by 1.
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So that'll be x to the n minus
1, the same power minus 1, one
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power below.
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So that's a highly important
formula.
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And actually it's important
to know it, not--
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right now, well, we've done
two or three examples.
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I guess the right way for me
to get this for n equals--
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so we really could check
1, 2, 3, and so on.
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All the positive integers.
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We could complete the proof.
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We could establish
the pattern.
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Actually, induction would
be one way to do it.
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If we know it for, as we did
here, for n equals 3, then
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we've got it for 4.
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If we know it for 4, the same
product formula would get it
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for 5 and onwards, and would
give us that answer.
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Good.
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Even better is the fact that
this formula is also true if n
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is a fraction.
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If we're doing the square root
of x, you recognize the square
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root of x is x to the--
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what's the exponent there
for square root?
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1/2.
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So I would like to
know for 1/2.
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OK, let me take a couple of
steps to get to that one.
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All right.
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The steps I'm going to take are
going to look just like
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this, but this was powers of x,
and it'll be very handy if
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I can do powers of f of x.
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I'd like to know--
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I want to find--
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So here's what I'm headed for.
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I'd like to know the derivative
of f of x to the
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n-th power equals what?
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That's what I'd like to know.
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So let me do f of x.
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Let me do it just
as I did before.
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Take n equals 2,
f of x squared.
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So what's the derivative of
f of x squared, like sine
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squared or whatever
we're squaring.
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Cosine squared.
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Well, for f of x squared, all
I'm doing is I'm taking f to
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be the same as g.
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I'll use the product rule.
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If g and f are the same, then
I've got something squared.
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And my product rule says that
the derivative-- and I just
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copy this rule.
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Now I'm taking p is going
to be f squared, right?
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Can I just write f
squared equals--
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so it's f times--
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f is the same as g.
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Are you with me?
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I'm just using the rule in a
very special case when the two
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functions are the same.
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The derivative of
f squared is f.
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What do I have? f times the
derivative of f, df dx.
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That's the first term.
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And then what's the
second term?
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Notice I wrote f instead of g,
because they're the same.
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And the second term is,
again, a copy of that.
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So I have 2 of these.
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Times 2, just the way
I had a 2 up there.
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This was the case
of x squared.
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This is the case of
f of x squared.
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Let me go one more
step to f cubed.
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What am I going to
do for f cubed?
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The derivative of--
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hold on.
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I have to show you what to
pay attention to here.
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To pay attention to is--
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the 2 we're familiar with.
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This would have been the x,
that's not a big deal.
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But there's something new.
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A df dx factor is coming in.
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It's going to stay with us.
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Let me see it here.
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The derivative of
f of x cubed.
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Now let's practice
with this one.
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OK.
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So now what am I
going to take?
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How do I get f of x cubed?
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Well, I've got f, so I'd better
take g to be f squared.
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Then when I multiply,
I've got cubed.
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So g is now going to be f
squared for this case.
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Can I take my product rule
with f times f squared?
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My product rule of f times
f squared is--
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I'm doing this now with g equals
f squared, just the way
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I did it over there at
some point with one
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of them as a square.
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OK.
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I'm near the end of
this calculation.
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OK.
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So what do I have. If this
thing is cubed, I
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have f times f squared.
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That's f cubed.
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And I take its derivative
by the rule.
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So I take f times the derivative
of f squared, which
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I just figured out
as 2f df dx.
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That's the f dg dx.
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And now I have g, which is
f squared, times df dx.
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What are you seeing there?
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You're seeing--
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well, again, these combine.
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That's what's nice about
this example.
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Here I have one f squared df dx,
and here I have two more.
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That's, all together, three.
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So the total was 3 times
f squared times df dx.
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And let me write down what
that pattern is saying.
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Here it will be n.
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Because here it was a 2.
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Here it's going to be
2 plus 1-- that's 3.
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And now if I have the n-th
power, I'm expecting an n
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times the next lower power
of f, f to the n
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minus 1, times what?
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Times this guy that's hanging
around, df dx.
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That's my--
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you could call that
the power rule.
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The derivative of a power.
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This would be the power rule
for just x to the n-th, and
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this is the derivative of a
function of x to the n-th.
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There's something special
here that we're
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going to see more of.
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This will be, also, an example
of what's coming as maybe the
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most important rule,
the chain rule.
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And typical of it is that when
I take this derivative, I
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follow that same pattern--
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n, this thing, to one lower
power, but then the derivative
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of what's inside.
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Can I use those words?
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Because I'll use it again
for the chain rule.
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n times one lower
power, times the
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derivative of what's inside.
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And why do I want to
do such a thing?
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Because I'd like to find out
the derivative of the
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square root of x.
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OK.
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Can we do that?
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I want to use this, now.
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So I want to use this to find
the derivative of the
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square root of x.
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OK.
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So that will be my function.
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f of x will be the
square root of x.
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So this is a good example.
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That's x to the 1/2 power.
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What would I love
to have happen?
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I would like this formula to
continue with n equals 1/2,
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but no change in the formula.
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And that does happen.
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How can I do that?
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OK, well, square root of
x is what I'm tackling.
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The easy thing would
be, if I square
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that, I'll get x, right?
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The square of the square root.
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Well, square root
of x squared--
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so there's f of x.
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00:14:15,130 --> 00:14:18,720
I'm just going to use the fact
that the square root of x
259
00:14:18,720 --> 00:14:20,130
squared is x.
260
00:14:20,130 --> 00:14:23,260
261
00:14:23,260 --> 00:14:24,470
Such is mathematics.
262
00:14:24,470 --> 00:14:30,830
You can write down really
straightforward ideas, but it
263
00:14:30,830 --> 00:14:32,310
had to come from somewhere.
264
00:14:32,310 --> 00:14:33,670
And now what am I going to do?
265
00:14:33,670 --> 00:14:35,650
I'm going to take
the derivative.
266
00:14:35,650 --> 00:14:38,840
Well, the derivative on
the right side is a 1.
267
00:14:38,840 --> 00:14:41,110
The derivative of x is 1.
268
00:14:41,110 --> 00:14:44,020
What is the derivative of
that left-hand side?
269
00:14:44,020 --> 00:14:45,750
Well, that fits my pattern.
270
00:14:45,750 --> 00:14:48,840
You see, here is my
f of x, squared.
271
00:14:48,840 --> 00:14:50,940
And I had a little
formula for the
272
00:14:50,940 --> 00:14:52,990
derivative of f of x squared.
273
00:14:52,990 --> 00:14:57,620
So the derivative of this
is 2 times the thing
274
00:14:57,620 --> 00:14:59,280
to one lower power--
275
00:14:59,280 --> 00:15:02,820
square root of x just
to the first power--
276
00:15:02,820 --> 00:15:07,260
times the derivative of what's
inside, if you allow me to use
277
00:15:07,260 --> 00:15:08,210
those words.
278
00:15:08,210 --> 00:15:10,360
It's this, df dx.
279
00:15:10,360 --> 00:15:13,630
And that's of course what I
actually wanted, the square
280
00:15:13,630 --> 00:15:14,880
root of x, dx.
281
00:15:14,880 --> 00:15:22,020
282
00:15:22,020 --> 00:15:24,310
This lecture is not going
to have too many more
283
00:15:24,310 --> 00:15:29,230
calculations, but this
is a good one to see.
284
00:15:29,230 --> 00:15:31,120
That's clear.
285
00:15:31,120 --> 00:15:32,990
I take the derivative
of both sides.
286
00:15:32,990 --> 00:15:34,040
That's clear.
287
00:15:34,040 --> 00:15:35,860
This is the 2 square
root of x.
288
00:15:35,860 --> 00:15:40,780
And now I've got what I want,
as soon as I move these over
289
00:15:40,780 --> 00:15:41,720
to the other side.
290
00:15:41,720 --> 00:15:43,710
So I divide by that.
291
00:15:43,710 --> 00:15:48,270
Can I now just do that with an
eraser, or maybe just X it
292
00:15:48,270 --> 00:15:50,130
out, and put it here.
293
00:15:50,130 --> 00:15:53,920
1 over 2 square root of x.
294
00:15:53,920 --> 00:15:56,410
Am I seeing what I want
for the derivative of
295
00:15:56,410 --> 00:15:57,560
square root of x?
296
00:15:57,560 --> 00:15:59,910
I hope so.
297
00:15:59,910 --> 00:16:02,720
I'm certainly seeing the 1/2.
298
00:16:02,720 --> 00:16:04,490
So the 1/2--
299
00:16:04,490 --> 00:16:05,290
that's the n.
300
00:16:05,290 --> 00:16:06,980
It's supposed to show up here.
301
00:16:06,980 --> 00:16:09,380
And then what do I
look for here?
302
00:16:09,380 --> 00:16:13,770
One lower power than 1/2,
which will be x
303
00:16:13,770 --> 00:16:16,050
to the minus 1/2.
304
00:16:16,050 --> 00:16:17,290
And is that what I have?
305
00:16:17,290 --> 00:16:19,520
Yes.
306
00:16:19,520 --> 00:16:20,970
You see the 1/2.
307
00:16:20,970 --> 00:16:24,820
And that square root of x,
that's x to the 1/2, but it's
308
00:16:24,820 --> 00:16:26,410
down in the denominator.
309
00:16:26,410 --> 00:16:30,030
And things in the
denominator--
310
00:16:30,030 --> 00:16:35,150
the exponent for those,
there's a minus sign.
311
00:16:35,150 --> 00:16:36,620
We'll come back to that.
312
00:16:36,620 --> 00:16:40,490
That's a crucial fact, going
back to algebra.
313
00:16:40,490 --> 00:16:45,970
But, you know, calculus is
now using all that--
314
00:16:45,970 --> 00:16:48,230
I won't say stuff.
315
00:16:48,230 --> 00:16:53,710
All those good things
that we learned in
316
00:16:53,710 --> 00:16:57,410
algebra, like exponents.
317
00:16:57,410 --> 00:17:00,980
So that was a good example.
318
00:17:00,980 --> 00:17:03,670
OK.
319
00:17:03,670 --> 00:17:08,560
So my pattern held
for n equals 1/2.
320
00:17:08,560 --> 00:17:15,710
And maybe I'll just say that
it also would hold for cube
321
00:17:15,710 --> 00:17:23,660
roots, and any root,
and other powers.
322
00:17:23,660 --> 00:17:26,089
In other words, I get
this formula.
323
00:17:26,089 --> 00:17:31,610
This is the handy formula that
we're trying to get.
324
00:17:31,610 --> 00:17:38,150
We got it very directly for
positive whole numbers.
325
00:17:38,150 --> 00:17:44,250
Now I'm getting it for
n equals 1 over any--
326
00:17:44,250 --> 00:17:47,510
now I'm getting it for capital
Nth roots, like 1/2.
327
00:17:47,510 --> 00:17:51,340
328
00:17:51,340 --> 00:17:55,910
Then I could go on
to get it for--
329
00:17:55,910 --> 00:18:00,630
I could take then the n-th
power of the n-th root.
330
00:18:00,630 --> 00:18:05,540
I could even stretch this
to get up to m over n.
331
00:18:05,540 --> 00:18:09,510
Any fraction, I can get to.
332
00:18:09,510 --> 00:18:15,080
But I can't get to negative
exponents yet, because those
333
00:18:15,080 --> 00:18:16,380
are divisions.
334
00:18:16,380 --> 00:18:19,060
Negative exponent is a division,
and I'm going to
335
00:18:19,060 --> 00:18:20,820
need the quotient rule,
which is right
336
00:18:20,820 --> 00:18:22,840
now still a big blank.
337
00:18:22,840 --> 00:18:23,560
OK.
338
00:18:23,560 --> 00:18:26,360
Pause for a moment.
339
00:18:26,360 --> 00:18:29,120
We've used the product rule.
340
00:18:29,120 --> 00:18:30,720
I haven't explained
it, though.
341
00:18:30,720 --> 00:18:35,640
Let me, so, explain
the product rule.
342
00:18:35,640 --> 00:18:37,720
Where did it come from?
343
00:18:37,720 --> 00:18:44,080
I'm going back before the
examples, and before that
344
00:18:44,080 --> 00:18:48,140
board full of chalk, back to
that formula and just think,
345
00:18:48,140 --> 00:18:50,300
where did it come from?
346
00:18:50,300 --> 00:18:54,560
How did we find the derivative
of f times g,
347
00:18:54,560 --> 00:18:55,740
of the product p?
348
00:18:55,740 --> 00:19:00,600
So we needed delta p, right?
349
00:19:00,600 --> 00:19:02,940
And then I'm going to
divide by delta x.
350
00:19:02,940 --> 00:19:03,340
OK.
351
00:19:03,340 --> 00:19:05,140
So let me try to make--
352
00:19:05,140 --> 00:19:08,760
what's the delta p when p is--
353
00:19:08,760 --> 00:19:10,955
remember, p is f times g.
354
00:19:10,955 --> 00:19:14,850
355
00:19:14,850 --> 00:19:19,730
Thinking about f times g, maybe
let's make it visual.
356
00:19:19,730 --> 00:19:23,200
Let's make it like a rectangle,
where this side is
357
00:19:23,200 --> 00:19:27,880
f of x and this side
is g of x.
358
00:19:27,880 --> 00:19:33,740
Then this area is f
times g, right?
359
00:19:33,740 --> 00:19:35,420
The area of a rectangle.
360
00:19:35,420 --> 00:19:37,460
And that's our p.
361
00:19:37,460 --> 00:19:41,580
OK, that's sitting there at x.
362
00:19:41,580 --> 00:19:44,760
Now move it a little.
363
00:19:44,760 --> 00:19:47,290
Move x a little bit.
364
00:19:47,290 --> 00:19:52,930
Move x a little and figure out,
how much does p change?
365
00:19:52,930 --> 00:19:53,700
That's our goal.
366
00:19:53,700 --> 00:19:56,280
We need the change in p.
367
00:19:56,280 --> 00:20:02,925
If I move x by a little bit,
then f changes a little, by a
368
00:20:02,925 --> 00:20:05,650
little amount, delta f, right?
369
00:20:05,650 --> 00:20:11,100
And g changes a little, by
a little amount, delta g.
370
00:20:11,100 --> 00:20:13,860
You remember those deltas?
371
00:20:13,860 --> 00:20:15,380
So it's the change in f.
372
00:20:15,380 --> 00:20:20,070
There's a delta x in here.
373
00:20:20,070 --> 00:20:24,490
x is the starting point.
374
00:20:24,490 --> 00:20:26,520
It's the thing we
move a little.
375
00:20:26,520 --> 00:20:32,700
When we move x a little, by
delta x, f will move a little,
376
00:20:32,700 --> 00:20:35,000
g will move a little, and their
377
00:20:35,000 --> 00:20:36,640
product will move a little.
378
00:20:36,640 --> 00:20:40,300
And now, can you see, in the
picture, where is the product?
379
00:20:40,300 --> 00:20:44,130
Well, this is where
f moved to.
380
00:20:44,130 --> 00:20:45,550
This is where g moved to.
381
00:20:45,550 --> 00:20:51,840
The product is this,
that bigger area.
382
00:20:51,840 --> 00:20:53,540
So where is delta p?
383
00:20:53,540 --> 00:20:56,120
Where is the change between
the bigger area and the
384
00:20:56,120 --> 00:20:56,900
smaller area?
385
00:20:56,900 --> 00:20:58,100
It's this.
386
00:20:58,100 --> 00:21:03,630
I have to figure out, what's
that new area?
387
00:21:03,630 --> 00:21:07,240
The delta p is in here.
388
00:21:07,240 --> 00:21:09,460
OK, can you see what
that area--
389
00:21:09,460 --> 00:21:11,330
well, look, here's
the way to do it.
390
00:21:11,330 --> 00:21:14,540
Cut it up into little
three pieces.
391
00:21:14,540 --> 00:21:17,090
392
00:21:17,090 --> 00:21:20,460
Because now they're little
rectangles, and we know the
393
00:21:20,460 --> 00:21:20,942
area of rectangles.
394
00:21:20,942 --> 00:21:22,050
Right?
395
00:21:22,050 --> 00:21:23,760
So help me out here.
396
00:21:23,760 --> 00:21:26,850
What is the area of
that rectangle?
397
00:21:26,850 --> 00:21:33,120
Well, its base is f, and
its height is delta g.
398
00:21:33,120 --> 00:21:36,055
So that is f times delta g.
399
00:21:36,055 --> 00:21:38,630
400
00:21:38,630 --> 00:21:41,150
What about this one?
401
00:21:41,150 --> 00:21:46,140
That has height g and
base delta f.
402
00:21:46,140 --> 00:21:51,080
So here I'm seeing a g times
delta f, for that area.
403
00:21:51,080 --> 00:21:54,750
404
00:21:54,750 --> 00:22:02,400
And what about this little
corner piece?
405
00:22:02,400 --> 00:22:07,210
Well, its height is just delta
g, its width is delta f.
406
00:22:07,210 --> 00:22:10,970
This is delta g times delta f.
407
00:22:10,970 --> 00:22:16,600
408
00:22:16,600 --> 00:22:21,290
And it's going to disappear.
409
00:22:21,290 --> 00:22:25,580
This is like a perfect place
to recognize that an
410
00:22:25,580 --> 00:22:27,300
expression--
411
00:22:27,300 --> 00:22:30,570
that's sort of like
second order.
412
00:22:30,570 --> 00:22:33,240
Let me use words without
trying to
413
00:22:33,240 --> 00:22:35,460
pin them down perfectly.
414
00:22:35,460 --> 00:22:40,410
Here is a zero-order, an
f, a real number, times
415
00:22:40,410 --> 00:22:42,530
a small delta g.
416
00:22:42,530 --> 00:22:43,740
So that's first order.
417
00:22:43,740 --> 00:22:46,370
That's going to show up--
418
00:22:46,370 --> 00:22:47,620
you'll see it disappear.
419
00:22:47,620 --> 00:22:51,000
420
00:22:51,000 --> 00:22:53,890
These three pieces, remember,
were the delta p.
421
00:22:53,890 --> 00:22:55,490
So what have I got here?
422
00:22:55,490 --> 00:22:59,890
I've got this piece, f delta
g, and I'm always
423
00:22:59,890 --> 00:23:01,970
dividing by delta x.
424
00:23:01,970 --> 00:23:06,640
And then I have this piece,
which is the g times the delta
425
00:23:06,640 --> 00:23:09,530
f, and I divide by
the delta x.
426
00:23:09,530 --> 00:23:13,330
And then this piece that I'm
claiming I don't have to worry
427
00:23:13,330 --> 00:23:17,400
much about, because I divide
that by delta x.
428
00:23:17,400 --> 00:23:19,580
So that was the third piece.
429
00:23:19,580 --> 00:23:20,830
This is it, now.
430
00:23:20,830 --> 00:23:23,310
431
00:23:23,310 --> 00:23:30,770
The picture has led to the
algebra, the formula for delta
432
00:23:30,770 --> 00:23:36,090
p, the change in the product
divided by delta x.
433
00:23:36,090 --> 00:23:38,030
That's what calculus says--
434
00:23:38,030 --> 00:23:42,760
OK, look at that, and then
take the tricky step, the
435
00:23:42,760 --> 00:23:47,740
calculus step, which is let
delta x get smaller and
436
00:23:47,740 --> 00:23:51,680
smaller and smaller,
approaching 0.
437
00:23:51,680 --> 00:23:57,320
So what do those three terms
do as delta x gets smaller?
438
00:23:57,320 --> 00:24:00,980
Well, all the deltas
get smaller.
439
00:24:00,980 --> 00:24:08,430
So what happens to this term
as delta x goes to 0?
440
00:24:08,430 --> 00:24:13,920
As the change in x is just
tiny, tiny, tiny?
441
00:24:13,920 --> 00:24:19,970
That term is the one that gives
the delta g over delta
442
00:24:19,970 --> 00:24:25,510
x, in the limit when delta x
goes to 0, is that one, right?
443
00:24:25,510 --> 00:24:31,730
And this guy is giving my g.
444
00:24:31,730 --> 00:24:34,020
That ratio is familiar, df dx.
445
00:24:34,020 --> 00:24:40,370
You see, the cool thing about
splitting it into these pieces
446
00:24:40,370 --> 00:24:44,090
was that we got this piece
by itself, which was
447
00:24:44,090 --> 00:24:46,370
just the f delta g.
448
00:24:46,370 --> 00:24:48,425
And we know what that does.
449
00:24:48,425 --> 00:24:50,130
It goes here.
450
00:24:50,130 --> 00:24:51,930
And this piece--
451
00:24:51,930 --> 00:24:53,190
we know what that does.
452
00:24:53,190 --> 00:24:57,600
And now, what about
this dumb piece?
453
00:24:57,600 --> 00:25:00,890
Well, as delta x goes
to 0, this would go
454
00:25:00,890 --> 00:25:05,310
to df dx, all right.
455
00:25:05,310 --> 00:25:06,780
But what would delta g do?
456
00:25:06,780 --> 00:25:08,350
It'll go to 0.
457
00:25:08,350 --> 00:25:12,190
You see, we have two little
things divided by only one
458
00:25:12,190 --> 00:25:12,760
little thing.
459
00:25:12,760 --> 00:25:18,780
This ratio is sensible, it gives
df dx, but this ratio is
460
00:25:18,780 --> 00:25:19,980
going to 0.
461
00:25:19,980 --> 00:25:22,060
So forget it.
462
00:25:22,060 --> 00:25:27,650
And now the two pieces that we
have are the two pieces of the
463
00:25:27,650 --> 00:25:30,150
product rule.
464
00:25:30,150 --> 00:25:30,590
OK.
465
00:25:30,590 --> 00:25:38,520
Product rule sort of visually
makes sense.
466
00:25:38,520 --> 00:25:39,670
OK.
467
00:25:39,670 --> 00:25:42,960
I'm ready to go to the
quotient rule.
468
00:25:42,960 --> 00:25:46,690
OK, so how am I going to deal,
now, with a ratio of
469
00:25:46,690 --> 00:25:48,310
f divided by g?
470
00:25:48,310 --> 00:25:50,930
471
00:25:50,930 --> 00:25:51,140
OK.
472
00:25:51,140 --> 00:25:54,330
Let's put that on
a fourth board.
473
00:25:54,330 --> 00:26:01,910
How to deal then with the
ratio of f over g.
474
00:26:01,910 --> 00:26:08,230
Well, what I know is the
product rule, right?
475
00:26:08,230 --> 00:26:14,290
So let me multiply both sides
by g of x and get a product.
476
00:26:14,290 --> 00:26:18,580
477
00:26:18,580 --> 00:26:21,160
There, that looks better.
478
00:26:21,160 --> 00:26:24,950
Of course the part that I don't
know is in here, but
479
00:26:24,950 --> 00:26:26,450
just fire away.
480
00:26:26,450 --> 00:26:27,935
Take the derivative
of both sides.
481
00:26:27,935 --> 00:26:29,060
OK.
482
00:26:29,060 --> 00:26:33,810
The derivative of the left
side is df dx, of course.
483
00:26:33,810 --> 00:26:35,990
Now I can use the
product rule.
484
00:26:35,990 --> 00:26:39,730
It's g of x, dq dx.
485
00:26:39,730 --> 00:26:42,950
That's the very, very
thing I'm wanting.
486
00:26:42,950 --> 00:26:43,590
dq dx--
487
00:26:43,590 --> 00:26:47,860
that's my big empty space.
488
00:26:47,860 --> 00:26:49,770
That's going to be the
quotient rule.
489
00:26:49,770 --> 00:26:56,730
And then the second one
is q of x times dg dx.
490
00:26:56,730 --> 00:26:59,900
491
00:26:59,900 --> 00:27:03,040
That's the product rule
applied to this.
492
00:27:03,040 --> 00:27:04,360
Now I have it.
493
00:27:04,360 --> 00:27:06,060
I've got dq dx.
494
00:27:06,060 --> 00:27:09,440
Well, I've got to get
it by itself.
495
00:27:09,440 --> 00:27:12,010
I want to get dq dx by itself.
496
00:27:12,010 --> 00:27:16,200
So I'm going to move this
part over there.
497
00:27:16,200 --> 00:27:18,840
Let me, even, multiply
both sides--
498
00:27:18,840 --> 00:27:23,980
this q, of course, I recognize
as f times g.
499
00:27:23,980 --> 00:27:27,080
This is f of x times g of x.
500
00:27:27,080 --> 00:27:29,350
That's what q was.
501
00:27:29,350 --> 00:27:32,362
Now I'm going to--
502
00:27:32,362 --> 00:27:35,360
oh, was not.
503
00:27:35,360 --> 00:27:37,500
It was f of x over g of x.
504
00:27:37,500 --> 00:27:38,750
Good Lord.
505
00:27:38,750 --> 00:27:41,900
506
00:27:41,900 --> 00:27:43,900
You would never have allowed
me to go on.
507
00:27:43,900 --> 00:27:45,420
OK.
508
00:27:45,420 --> 00:27:46,270
Good.
509
00:27:46,270 --> 00:27:49,920
This is came from the product
rule, and now my final job is
510
00:27:49,920 --> 00:27:55,610
just to isolate dq dx and
see what I've got.
511
00:27:55,610 --> 00:27:59,820
What I'll have will be
the quotient rule.
512
00:27:59,820 --> 00:28:04,460
One good way is if I multiply
both sides by g.
513
00:28:04,460 --> 00:28:11,410
So I multiply everything by
g, so here's a g, df dx.
514
00:28:11,410 --> 00:28:14,890
And now this guy I'm going to
bring over to the other side.
515
00:28:14,890 --> 00:28:17,870
When I multiply that by g, that
just knocks that out.
516
00:28:17,870 --> 00:28:19,620
When I bring it over,
it comes over with a
517
00:28:19,620 --> 00:28:22,480
minus sign, f dg dx.
518
00:28:22,480 --> 00:28:26,440
519
00:28:26,440 --> 00:28:31,040
And this one got multiplied by
g, so right now I'm looking at
520
00:28:31,040 --> 00:28:36,830
g squared, dq dx.
521
00:28:36,830 --> 00:28:38,080
The guy I want.
522
00:28:38,080 --> 00:28:44,190
523
00:28:44,190 --> 00:28:46,850
Again, just algebra.
524
00:28:46,850 --> 00:28:49,170
Moving stuff from one
side to the other
525
00:28:49,170 --> 00:28:50,940
produced the minus sign.
526
00:28:50,940 --> 00:28:54,310
Multiplying by g, you
see what happened.
527
00:28:54,310 --> 00:28:56,420
So what do I now finally do?
528
00:28:56,420 --> 00:28:58,380
I'm ready to write
this formula in.
529
00:28:58,380 --> 00:29:02,220
530
00:29:02,220 --> 00:29:03,380
I've got it there.
531
00:29:03,380 --> 00:29:06,690
I've got dq dx, just
as soon as I divide
532
00:29:06,690 --> 00:29:10,410
both sides by g squared.
533
00:29:10,410 --> 00:29:12,470
So let me write that
left-hand side.
534
00:29:12,470 --> 00:29:23,910
g df dx minus f dg dx, and I
have to divide everything--
535
00:29:23,910 --> 00:29:26,240
this g squared has got
to come down here.
536
00:29:26,240 --> 00:29:28,060
It's a little bit messier
formula but
537
00:29:28,060 --> 00:29:29,540
you get used to it.
538
00:29:29,540 --> 00:29:30,790
g squared.
539
00:29:30,790 --> 00:29:34,210
540
00:29:34,210 --> 00:29:37,380
That's the quotient rule.
541
00:29:37,380 --> 00:29:38,750
Can I say it in words?
542
00:29:38,750 --> 00:29:41,390
Because I actually say
those words to myself
543
00:29:41,390 --> 00:29:46,680
every time I use it.
544
00:29:46,680 --> 00:29:50,790
So here are the words I say,
because that's a kind of
545
00:29:50,790 --> 00:29:52,060
messy-looking expression.
546
00:29:52,060 --> 00:29:54,270
But if you just think
about words--
547
00:29:54,270 --> 00:29:59,280
so for me, remember we're
dealing with f over g. f is
548
00:29:59,280 --> 00:30:01,170
the top, g at the bottom.
549
00:30:01,170 --> 00:30:04,830
So I say to myself, the bottom
times the derivative of the
550
00:30:04,830 --> 00:30:10,160
top minus the top times the
derivative of the bottom,
551
00:30:10,160 --> 00:30:14,630
divided by the bottom squared.
552
00:30:14,630 --> 00:30:16,910
That wasn't brilliant,
but anyway, I
553
00:30:16,910 --> 00:30:18,900
remember it that way.
554
00:30:18,900 --> 00:30:19,730
OK.
555
00:30:19,730 --> 00:30:25,645
so now, finally, I'm ready to go
further with this pattern.
556
00:30:25,645 --> 00:30:29,500
557
00:30:29,500 --> 00:30:33,010
I still like that pattern.
558
00:30:33,010 --> 00:30:37,330
We've got the quotient rule, so
the two rules are now set,
559
00:30:37,330 --> 00:30:40,810
and I want to do one last
example before stopping.
560
00:30:40,810 --> 00:30:44,550
And that example is going to
be a quotient, of course.
561
00:30:44,550 --> 00:30:52,330
And it might as well be
a negative power of x.
562
00:30:52,330 --> 00:30:56,650
So now my example--
563
00:30:56,650 --> 00:30:59,460
last example for today--
564
00:30:59,460 --> 00:31:03,690
my quotient is going to be 1.
565
00:31:03,690 --> 00:31:09,380
The f of x will be 1 and the
g of x-- so this is my f.
566
00:31:09,380 --> 00:31:11,360
This is my g.
567
00:31:11,360 --> 00:31:13,220
I have a ratio of two things.
568
00:31:13,220 --> 00:31:21,285
And as I've said, this
is x to the minus n.
569
00:31:21,285 --> 00:31:23,570
Right?
570
00:31:23,570 --> 00:31:25,650
That's what we mean.
571
00:31:25,650 --> 00:31:29,800
We can think again
about exponents.
572
00:31:29,800 --> 00:31:37,520
A negative exponent becomes
positive when it's in the
573
00:31:37,520 --> 00:31:38,930
denominator.
574
00:31:38,930 --> 00:31:43,880
And we want it in the
denominator so we can use this
575
00:31:43,880 --> 00:31:45,820
crazy quotient rule.
576
00:31:45,820 --> 00:31:46,280
All right.
577
00:31:46,280 --> 00:31:49,120
So let me think through
the quotient rule.
578
00:31:49,120 --> 00:31:58,580
So the derivative of this ratio,
which is x to the minus
579
00:31:58,580 --> 00:32:02,750
n That's the q, is 1
over x to the n.
580
00:32:02,750 --> 00:32:04,260
The derivative is--
581
00:32:04,260 --> 00:32:06,800
OK, ready for the
quotient rule?
582
00:32:06,800 --> 00:32:11,950
Bottom times the derivative
of the top--
583
00:32:11,950 --> 00:32:14,350
ah, but the top's just
a constant, so its
584
00:32:14,350 --> 00:32:16,350
derivative is 0--
585
00:32:16,350 --> 00:32:17,390
minus--
586
00:32:17,390 --> 00:32:21,210
remembering that minus-- the
top times the derivative of
587
00:32:21,210 --> 00:32:22,460
the bottom.
588
00:32:22,460 --> 00:32:24,260
589
00:32:24,260 --> 00:32:24,445
Ha.
590
00:32:24,445 --> 00:32:27,190
Now we have a chance to use our
591
00:32:27,190 --> 00:32:30,550
pattern with a plus exponent.
592
00:32:30,550 --> 00:32:36,620
The derivative of the bottom
is nx to the n minus 1.
593
00:32:36,620 --> 00:32:40,230
So it's two terms, again,
but with a minus sign.
594
00:32:40,230 --> 00:32:44,830
And then the other thing I must
remember is, divide by g
595
00:32:44,830 --> 00:32:51,520
squared, x to the
n twice squared.
596
00:32:51,520 --> 00:32:52,790
OK.
597
00:32:52,790 --> 00:32:54,730
That's it.
598
00:32:54,730 --> 00:32:57,440
Of course, I'm going
to simplify it,
599
00:32:57,440 --> 00:32:59,430
and then I'm done.
600
00:32:59,430 --> 00:33:01,280
So this is 0.
601
00:33:01,280 --> 00:33:02,680
Gone.
602
00:33:02,680 --> 00:33:07,170
This is minus n, which I like.
603
00:33:07,170 --> 00:33:09,690
I like to see minus
n come down.
604
00:33:09,690 --> 00:33:12,620
That's my pattern, that this
exponent should come down.
605
00:33:12,620 --> 00:33:15,450
Minus n, and then I want
to see-- oh, what
606
00:33:15,450 --> 00:33:16,480
else do I have here?
607
00:33:16,480 --> 00:33:18,050
What's the power of x?
608
00:33:18,050 --> 00:33:20,440
Well, here I have an
x to the n-th.
609
00:33:20,440 --> 00:33:25,930
And here I have, twice, so can
I cancel this one and just
610
00:33:25,930 --> 00:33:28,540
keep this one?
611
00:33:28,540 --> 00:33:30,930
So I still have an
x to the minus 1.
612
00:33:30,930 --> 00:33:33,530
I don't let him go.
613
00:33:33,530 --> 00:33:36,430
Actually the pattern's here.
614
00:33:36,430 --> 00:33:43,980
The answer is minus n minus
capital N, which was the
615
00:33:43,980 --> 00:33:49,700
exponent, times x to
one smaller power.
616
00:33:49,700 --> 00:33:52,690
This is x to the minus n, and
then there's another x
617
00:33:52,690 --> 00:33:54,060
to the minus 1.
618
00:33:54,060 --> 00:34:01,770
The final result was that the
derivative is minus nx to the
619
00:34:01,770 --> 00:34:04,950
minus n, minus 1.
620
00:34:04,950 --> 00:34:11,360
And that's the good pattern
that matches here.
621
00:34:11,360 --> 00:34:17,460
When little n matches minus
big N, that pattern is the
622
00:34:17,460 --> 00:34:18,199
same as that.
623
00:34:18,199 --> 00:34:24,860
So we now have the derivatives
of powers of x as an example
624
00:34:24,860 --> 00:34:28,830
from the quotient rule
and the product rule.
625
00:34:28,830 --> 00:34:30,407
Well, I just have to
say one thing.
626
00:34:30,407 --> 00:34:31,920
We haven't got--
627
00:34:31,920 --> 00:34:38,820
We've fractions, we've got
negative numbers, but we don't
628
00:34:38,820 --> 00:34:42,469
have a whole lot of other
numbers, like pi.
629
00:34:42,469 --> 00:34:45,989
We don't know what is,
for example, the
630
00:34:45,989 --> 00:34:49,110
derivative of x to the pi.
631
00:34:49,110 --> 00:34:51,150
Because pi isn't--
632
00:34:51,150 --> 00:34:54,909
pi is positive, so we're OK in
the product rule, but it's not
633
00:34:54,909 --> 00:34:56,820
a fraction and we haven't
got it yet.
634
00:34:56,820 --> 00:34:59,370
635
00:34:59,370 --> 00:35:02,400
What do you think it is?
636
00:35:02,400 --> 00:35:03,490
You're right--
637
00:35:03,490 --> 00:35:09,520
it is pi x to the pi minus 1.
638
00:35:09,520 --> 00:35:13,340
Well, actually I never met x
to the pi in my life, until
639
00:35:13,340 --> 00:35:21,340
just there, but I've certainly
met all kinds of powers of x
640
00:35:21,340 --> 00:35:25,310
and this is just one
more example.
641
00:35:25,310 --> 00:35:25,350
OK.
642
00:35:25,350 --> 00:35:26,600
So that's quotient rule--
643
00:35:26,600 --> 00:35:28,150
644
00:35:28,150 --> 00:35:36,270
first came product rule, power
rule, and then quotient rule,
645
00:35:36,270 --> 00:35:39,770
leading to this calculation.
646
00:35:39,770 --> 00:35:42,200
Now, the quotient rule I can
use for other things, like
647
00:35:42,200 --> 00:35:46,330
sine x over cosine x.
648
00:35:46,330 --> 00:35:50,400
We're far along, and one
more big rule will
649
00:35:50,400 --> 00:35:52,440
be the chain rule.
650
00:35:52,440 --> 00:35:54,770
OK, that's for another time.
651
00:35:54,770 --> 00:35:55,976
Thank you.
652
00:35:55,976 --> 00:35:56,410
[NARRATOR:]
653
00:35:56,410 --> 00:35:57,770
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654
00:35:57,770 --> 00:36:00,600
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655
00:36:00,600 --> 00:36:02,870
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656
00:36:02,870 --> 00:36:04,090
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657
00:36:04,090 --> 00:36:07,220
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658
00:36:07,220 --> 00:36:10,300
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659
00:36:10,300 --> 00:36:11,860
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660
00:36:11,860 --> 00:36:13,990