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PROFESSOR: OK, earlier lecture
introduced the logarithm as

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the inverse function
to the exponential.

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And now it's time to do
calculus, find its derivative.

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And we spoke about other inverse
functions, and here is

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an important one: the inverse
sine function, or sometimes

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called the arc sine.

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We'll find its derivative,
too.

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OK, so what we're doing really
is sort of completing the list

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of important rules
for derivatives.

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We know about the derivative
of a sum.

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Just add the derivatives.

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f minus g, just subtract
the derivatives.

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We know the product rule and the
quotient rule, so that's

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add, subtract, multiply,
divide functions.

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And then very, very important is
the chain of functions, the

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chain rule.

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This is never to be mixed
up with that.

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You wouldn't do such a thing.

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And now we're adding
one more f inverse.

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That's today, the derivative
of the inverse.

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That will really complete
the rules.

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Out of simple functions like
exponential, sine and cosine,

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powers of x, this creates all
the rest of the functions that

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we typically use.

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OK, so let's start with the most
important of all: the f

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of x is e to the x.

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And then the inverse
function, we

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named the natural logarithm.

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And notice, remember how
I reversed the letters.

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Here, x is the input and y
is the output from the

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exponential function.

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So for the inverse function,
y is the input.

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I go backwards to x, and the
thing to remember, the one

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thing to remember, just tell
yourself x is the exponent.

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The logarithm is the exponent.

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OK, so a chain of functions
is coming here, and it's a

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perfectly terrific chain.

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This is the rule for
inverse functions.

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If I start with an x and I do
f of x, that gets me to y.

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And now I do the inverse
function, and it

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brings me back to x.

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So I really have a chain
of functions.

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The chain has a very
special result.

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And our situation is if we
know how to take the

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derivative of f, this ought to
tell us-- the chain rule--

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how to take the derivative
of the

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inverse function, f inverse.

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Let me try it with this
all-important example.

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So which--

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and notice also, chain
goes the other way.

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If I start with y, do the
inverse function, then

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I've reached x.

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Then f of x is y.

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Maybe before I take e to the
x, let me take the function

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that always is the
starting point.

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So practice is--

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I'll call this example of--

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just to remember how inverse
functions work--

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linear functions.

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y equals ax plus b.

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That's f of x there.

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Linear.

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What's the inverse of that?

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Now the point about inverses
is I want to solve for x in

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terms of y.

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I want to get x by itself.

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So I move b to the
opposite side.

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So in the end, I want to
get x equals something.

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And how do I do that?

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I move b to the opposite side,
and then I still have ax, so I

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divide by a.

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Then I've got x by itself.

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This is the inverse.

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This is f inverse of y.

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Notice something about inverse
functions, that here we did--

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this function f of x was
created in two steps.

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it was sort of a chain
in itself.

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The first step was
multiply by a.

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We multiply and then we add.

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What's inverse function do?

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The inverse function takes y.

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Subtracts b.

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So it does subtract first
to get y minus

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b, and then it divides.

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What's my point?

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My point is that if a function
is built in two steps,

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multiply and then add in this
nice case, the inverse

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function does the
inverse steps.

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Instead of multiplying
it divides.

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Instead of add, it subtracts.

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What it does though is
the opposite order.

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Notice the multiply was done
first, then the divide is

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last. The add was done second,
the subtract was done first.

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When you invert things, the
order-- well, you know it.

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It has to be that way.

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It's that way in life, right?

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If we're standing on the beach
and we walk to the water and

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then we swim to the dock, so
that's our function f of x

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from where we were to the dock,
then how do we get back?

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Well, wise to swim
first, right?

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You don't want to walk first.
From the dock, you swim back.

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So you swam one way at the
end, and then in the

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inverse, you swim--

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oh, you get it.

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OK, so now I'm ready
for the real thing.

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I'll take one of those chains
and take its derivative.

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Let me take the first one.

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So the first one says that the
log of e to the x is x.

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That's--

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The logarithm was defined
that way.

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That's what the log is.

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Let's take the derivative of
that equation, the derivative

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of both sides.

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We will be learning what's the
derivative of the log.

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That's what we don't know.

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So if I take the derivative,
well, on the right-hand side,

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I certainly get 1.

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On the left-hand side, this
is where it's interesting.

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So it's the chain rule.

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The log of something, so I
should take the derivative--

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oh, I need a little more
space here, over here.

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The derivative of log
y, y is e to the x.

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Everybody's got that, right?

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So this is log y, and I'm taking
its derivative with

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respect to y.

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But then I have to, as the chain
rule tells me, take the

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derivative of what's inside.

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What's inside is e to the
x, so I have dy dx.

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So I put dy dx.

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And the derivative on the
right-hand side, the neat

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point here is that the
x-derivative of that is 1.

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OK, now I'm going to learn what
this is because I know

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what this is: the derivative
of dy dx, the derivative of

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the exponential.

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Well, now comes the most
important property that we use

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to construct this exponential.

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dy dx is e to the x.

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No problem, OK?

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And now, I'm going to divide
by it to get what I want--

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almost. Almost, I say.

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Well, I've got the derivative
of log y here.

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Correct, but there's a
step to take still.

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I have to write--

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I want a function of y.

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The log is a function of y.

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It's derivative is--

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the answer is a function of y.

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So I have to go back
from x to y.

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But that's simple.

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e to the x is y.

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Oh!

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Look at this fantastic answer.

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The derivative of the log, the
thing we wanted, is 1 over y.

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Why do I say fantastic?

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Because out of the blue almost,
we've discovered the

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function log y, which has that
derivative 1 over y.

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And the point is this is
the minus 1 power.

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It's the only power that we
didn't produce earlier as a

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derivative.

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I have to make that point.

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You remember the very first
derivatives we knew were the

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derivatives of x to the
n-th, powers of x.

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Everybody knows that that's n
times x to the n minus 1.

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The derivative of every power
is one power below.

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With one exception.

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With one exception.

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If n is 0, so I have to
put except n equals 0.

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Well, it's true when n is 0,
so I don't mean the formula

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doesn't fail.

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What fails is when n is 0, this
right-hand side is 0, and

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I don't get the minus 1 power.

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No power of x produces the minus
1 power when I take the

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derivative.

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So that was like an open hole
in the list of derivatives.

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Nobody was giving the derivative
to be the minus 1

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power when we were looking
at the powers of x.

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Well, here it showed up.

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00:10:47,790 --> 00:10:51,890

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Now, you'll say the
letter y's there.

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OK, that's the 25th letter
of the alphabet.

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I'm perfectly happy if you
prefer the 26th letter.

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You can write d log z dz equals
1/z if you want to.

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You can write, as you might
like to, d by dx.

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Use the 24th letter
of log x is 1/x.

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00:11:15,550 --> 00:11:18,330

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00:11:18,330 --> 00:11:24,820
I'm perfectly OK for you to do
that, to write the x there,

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now after we got the formula.

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Up to this point, I really had
to keep x and y straight

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because I was beginning
from y is e to the x.

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That was my starting point.

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OK, so that keeping them
straight got me the derivative

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of log y as 1/y.

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00:11:49,607 --> 00:11:51,040
End.

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00:11:51,040 --> 00:11:54,360
Now, I'm totally happy if you
use any other letter.

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00:11:54,360 --> 00:11:56,970
Use t if you have
things growing.

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And remember about the
logarithm now.

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We can see why it
grows so slowly.

202
00:12:06,240 --> 00:12:10,210
Because its slope is 1/y.

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00:12:10,210 --> 00:12:13,870
Or let's look at this one,
because we're used to thinking

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of graphs with x
along the axis.

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00:12:20,400 --> 00:12:23,510
And this is telling us that the
slope of the log curve--

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the log curve is increasing, but
the slope is decreasing,

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getting smaller and smaller.

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00:12:30,170 --> 00:12:35,290
As x gets very small, it's
just barely increasing.

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00:12:35,290 --> 00:12:38,220
It does keep going on
up to infinity,

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but very, very slowly.

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00:12:40,490 --> 00:12:41,470
And why is that?

212
00:12:41,470 --> 00:12:46,350
That's because the exponential
is going very, very quickly.

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00:12:46,350 --> 00:12:49,740
And you remember that the one
graph is just the flip of the

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00:12:49,740 --> 00:12:53,060
other graph, so if one is
climbing like mad, the other

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00:12:53,060 --> 00:12:56,420
one is growing slowly.

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00:12:56,420 --> 00:13:03,970
OK, that's the main facts, the
most important formula of

217
00:13:03,970 --> 00:13:05,250
today's lecture.

218
00:13:05,250 --> 00:13:07,960
I could--

219
00:13:07,960 --> 00:13:11,310
do you feel like practice to
take the chain in the opposite

220
00:13:11,310 --> 00:13:16,780
direction just to see
what would happen?

221
00:13:16,780 --> 00:13:20,560
So what's the opposite
direction?

222
00:13:20,560 --> 00:13:23,740
I guess the opposite direction
is to start with--

223
00:13:23,740 --> 00:13:24,990
which did I start with?

224
00:13:24,990 --> 00:13:29,060
I started with log of
e to the x is x.

225
00:13:29,060 --> 00:13:33,590
The opposite direction would be
to start with e to the log

226
00:13:33,590 --> 00:13:38,530
y is y, right?

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00:13:38,530 --> 00:13:39,910
That's the same chain.

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00:13:39,910 --> 00:13:43,750
That's the f inverse coming
before the f.

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00:13:43,750 --> 00:13:45,990
What do I do?

230
00:13:45,990 --> 00:13:47,900
Take derivatives.

231
00:13:47,900 --> 00:13:50,540
Take the derivative
of everything, OK?

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00:13:50,540 --> 00:13:56,040
So take the derivative,
the y-derivative.

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00:13:56,040 --> 00:13:57,220
I get the nice thing.

234
00:13:57,220 --> 00:14:00,330
I mean, that's the fun part,
taking the derivative on the

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right-hand side.

236
00:14:01,650 --> 00:14:05,440
On the left side, a little more
work, but I know how to

237
00:14:05,440 --> 00:14:09,340
take the derivative of
e to the something.

238
00:14:09,340 --> 00:14:10,750
It's the chain rule.

239
00:14:10,750 --> 00:14:12,370
Of course it's the chain rule.

240
00:14:12,370 --> 00:14:14,050
We got a chain here.

241
00:14:14,050 --> 00:14:16,940
So the derivative of e to the
something, now you remember

242
00:14:16,940 --> 00:14:25,110
with the chain rule, is e to
that same something times the

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00:14:25,110 --> 00:14:27,810
derivative of what's inside.

244
00:14:27,810 --> 00:14:30,950
The derivative and what's
inside is this guy: the

245
00:14:30,950 --> 00:14:34,180
derivative of log y dy.

246
00:14:34,180 --> 00:14:36,780

247
00:14:36,780 --> 00:14:41,340
This is what we want to know,
the one we know, and what is e

248
00:14:41,340 --> 00:14:42,590
to the log y?

249
00:14:42,590 --> 00:14:45,430

250
00:14:45,430 --> 00:14:48,190
It's sitting up there
on the line before.

251
00:14:48,190 --> 00:14:50,440
e to the log y is y.

252
00:14:50,440 --> 00:14:54,340
So this parenthesis is
just containing y.

253
00:14:54,340 --> 00:14:55,670
Bring it down.

254
00:14:55,670 --> 00:14:58,270
Set it under there, and
you have it again.

255
00:14:58,270 --> 00:15:05,170
The derivative of log y
dy is 1 over e to the

256
00:15:05,170 --> 00:15:06,660
log y, which is y.

257
00:15:06,660 --> 00:15:09,800

258
00:15:09,800 --> 00:15:14,430
OK, we sort of have done more
about inverse functions than

259
00:15:14,430 --> 00:15:20,600
typical lectures might, but I
did it really because they're

260
00:15:20,600 --> 00:15:23,910
kind of not so simple.

261
00:15:23,910 --> 00:15:30,150
And yet, they're crucially
important in this situation of

262
00:15:30,150 --> 00:15:32,240
connecting exponential
with log.

263
00:15:32,240 --> 00:15:35,556
And by the way, I prefer to
start with exponential.

264
00:15:35,556 --> 00:15:38,480

265
00:15:38,480 --> 00:15:41,180
The logic goes also just fine.

266
00:15:41,180 --> 00:15:45,740
In fact, some steps are a little
smoother if you start

267
00:15:45,740 --> 00:15:51,720
with a logarithm function,
define that somehow, and then

268
00:15:51,720 --> 00:15:55,810
take its inverse, which will
be the exponential.

269
00:15:55,810 --> 00:16:01,680
But for me, the exponential
is so all important.

270
00:16:01,680 --> 00:16:05,150
The logarithm is important, but
it's not in the league of

271
00:16:05,150 --> 00:16:06,260
e to the x.

272
00:16:06,260 --> 00:16:11,640
So I prefer to do it this
way to know e to the x.

273
00:16:11,640 --> 00:16:15,530
Now if you bear with me,
I'll do the other

274
00:16:15,530 --> 00:16:17,480
derivative for today.

275
00:16:17,480 --> 00:16:21,220
The other derivative
is this one.

276
00:16:21,220 --> 00:16:24,470
Can we do that?

277
00:16:24,470 --> 00:16:29,520
OK, so I want the derivative
of this arc sine

278
00:16:29,520 --> 00:16:32,610
function, all right?

279
00:16:32,610 --> 00:16:34,230
So I'm going to--

280
00:16:34,230 --> 00:16:36,850
let me bring that.

281
00:16:36,850 --> 00:16:42,150
This side of the board is now
going to be x is the inverse

282
00:16:42,150 --> 00:16:49,810
sine of y, or it's often called
the arc sine of y.

283
00:16:49,810 --> 00:16:51,060
OK, good.

284
00:16:51,060 --> 00:16:54,940

285
00:16:54,940 --> 00:16:56,190
All right.

286
00:16:56,190 --> 00:17:02,150

287
00:17:02,150 --> 00:17:06,579
So again, I have a chain.

288
00:17:06,579 --> 00:17:08,020
I start with x.

289
00:17:08,020 --> 00:17:10,369
I create y.

290
00:17:10,369 --> 00:17:12,560
So y is sine x.

291
00:17:12,560 --> 00:17:24,569
So y is the sine of x, but
x is the arc sine of y.

292
00:17:24,569 --> 00:17:25,670
That's the chain.

293
00:17:25,670 --> 00:17:26,810
Start with a y.

294
00:17:26,810 --> 00:17:27,760
Do f inverse.

295
00:17:27,760 --> 00:17:30,810
Do f, and you got y
again, all right?

296
00:17:30,810 --> 00:17:34,820
Now, I'm interested in the
derivative, the derivative of

297
00:17:34,820 --> 00:17:36,270
this arc sine of y.

298
00:17:36,270 --> 00:17:37,530
I want the y-derivative.

299
00:17:37,530 --> 00:17:43,360
I'm just going to copy this
plan, but instead of e, I've

300
00:17:43,360 --> 00:17:44,630
got sines here.

301
00:17:44,630 --> 00:17:48,860
So take the y-derivative of both
sides, the y-derivative

302
00:17:48,860 --> 00:17:50,170
of both sides.

303
00:17:50,170 --> 00:17:53,120
Well, I always like that one.

304
00:17:53,120 --> 00:17:56,250
The y-derivative of this
is the chain rule.

305
00:17:56,250 --> 00:17:59,560
So I have the sine of some
inside function.

306
00:17:59,560 --> 00:18:06,890
So the derivative is the cosine
of that inside function

307
00:18:06,890 --> 00:18:11,620
times the derivative of the
inside function, which is the

308
00:18:11,620 --> 00:18:12,870
guy we want.

309
00:18:12,870 --> 00:18:16,260

310
00:18:16,260 --> 00:18:24,500
OK, so I have to figure
out that thing.

311
00:18:24,500 --> 00:18:28,500
In other words, I guess I've
got to think a little bit

312
00:18:28,500 --> 00:18:32,880
about these inverse
trig functions.

313
00:18:32,880 --> 00:18:38,560
OK, so what's the story with
the inverse trig functions?

314
00:18:38,560 --> 00:18:41,740
The point will be this
is an angle.

315
00:18:41,740 --> 00:18:43,000
Ha!

316
00:18:43,000 --> 00:18:44,020
That's an angle.

317
00:18:44,020 --> 00:18:47,450
Let me draw the triangle.

318
00:18:47,450 --> 00:18:50,250
Here is my angle theta.

319
00:18:50,250 --> 00:18:53,870
Here is my sine theta.

320
00:18:53,870 --> 00:18:58,590
Here is my cos theta, and
everybody knows that now the

321
00:18:58,590 --> 00:19:00,020
hypotenuse is 1.

322
00:19:00,020 --> 00:19:03,410

323
00:19:03,410 --> 00:19:04,660
So here is theta.

324
00:19:04,660 --> 00:19:07,170

325
00:19:07,170 --> 00:19:08,280
OK, whoa!

326
00:19:08,280 --> 00:19:09,060
Wait a minute.

327
00:19:09,060 --> 00:19:13,330
I would love theta to
be the angle whose--

328
00:19:13,330 --> 00:19:14,900
oh, maybe it is.

329
00:19:14,900 --> 00:19:17,430
This is the angle whose sine--

330
00:19:17,430 --> 00:19:25,220
theta should be the angle
whose sine is y, right?

331
00:19:25,220 --> 00:19:30,670
OK, theta is the angle
whose sine is y.

332
00:19:30,670 --> 00:19:33,040
OK, let me make that happen.

333
00:19:33,040 --> 00:19:37,440

334
00:19:37,440 --> 00:19:41,380
And now, tell me the other side
because I got to get a

335
00:19:41,380 --> 00:19:42,880
cosine in here somewhere.

336
00:19:42,880 --> 00:19:45,740
What is this side?

337
00:19:45,740 --> 00:19:49,380
Back to Pythagoras, the
most important fact

338
00:19:49,380 --> 00:19:50,960
about a right triangle.

339
00:19:50,960 --> 00:19:53,560
This side will be the
square root of--

340
00:19:53,560 --> 00:19:57,110
this squared plus this squared
is 1, so this is the square

341
00:19:57,110 --> 00:20:01,790
root of 1 minus y squared.

342
00:20:01,790 --> 00:20:04,510
And that's the cosine.

343
00:20:04,510 --> 00:20:12,635
The cosine of this angle theta
is this guy divided by 1.

344
00:20:12,635 --> 00:20:16,970

345
00:20:16,970 --> 00:20:23,800
We're there, and all I've used
pretty quickly was I popped up

346
00:20:23,800 --> 00:20:24,980
a triangle there.

347
00:20:24,980 --> 00:20:27,060
I named an angle theta.

348
00:20:27,060 --> 00:20:30,760
I took its sine to be y, and
I figured out what its

349
00:20:30,760 --> 00:20:32,110
cosine had to be.

350
00:20:32,110 --> 00:20:34,590
OK, so there's the theta.

351
00:20:34,590 --> 00:20:40,720
Its cosine has to be this,
and now I'm ready to

352
00:20:40,720 --> 00:20:43,290
write out the answer.

353
00:20:43,290 --> 00:20:45,480
I'm ready to write down
the answer there.

354
00:20:45,480 --> 00:20:48,050
That has a 1 equals--

355
00:20:48,050 --> 00:20:55,010
the cosine of theta, that's this
times the derivative of

356
00:20:55,010 --> 00:20:56,840
the inverse sine.

357
00:20:56,840 --> 00:21:02,990

358
00:21:02,990 --> 00:21:15,360
You see, I had to get this
expression into something--

359
00:21:15,360 --> 00:21:16,750
I had to solve it for y.

360
00:21:16,750 --> 00:21:18,850
I had to figure out what
that quantity is as

361
00:21:18,850 --> 00:21:20,120
a function of y.

362
00:21:20,120 --> 00:21:23,060
And now I just put
this down below.

363
00:21:23,060 --> 00:21:28,080
So if I cross this out
and put it down here,

364
00:21:28,080 --> 00:21:30,770
I've got the answer.

365
00:21:30,770 --> 00:21:36,850
There is the derivative of the
arc sine function: 1 over the

366
00:21:36,850 --> 00:21:39,350
square root of 1 minus
y squared.

367
00:21:39,350 --> 00:21:51,670
OK, it's not as beautiful as
1/y, but it shows up in a lot

368
00:21:51,670 --> 00:21:56,570
of problems. As we said earlier,
sines and cosines are

369
00:21:56,570 --> 00:22:01,990
involved with repeated motion,
going around a circle, going

370
00:22:01,990 --> 00:22:05,780
up and down, going across
and back, in and out.

371
00:22:05,780 --> 00:22:10,310
And it will turn out that this
quantity, which is really

372
00:22:10,310 --> 00:22:14,690
coming from the Pythagoras, is
going to turn up, and we'll

373
00:22:14,690 --> 00:22:19,020
need to know that it's the
derivative of the arc sine.

374
00:22:19,020 --> 00:22:22,340
And may I just write down what's
the derivative of the

375
00:22:22,340 --> 00:22:24,490
arc cosine as long
as we're at it?

376
00:22:24,490 --> 00:22:25,960
And then I'm done.

377
00:22:25,960 --> 00:22:31,450
The derivative of the
arc cosine, well,

378
00:22:31,450 --> 00:22:32,800
you remember what--

379
00:22:32,800 --> 00:22:37,160
what's the difference between
sines and cosines when we take

380
00:22:37,160 --> 00:22:38,410
derivatives?

381
00:22:38,410 --> 00:22:42,230
The cosine has a minus.

382
00:22:42,230 --> 00:22:46,710
So there'll be a minus 1
over the square root

383
00:22:46,710 --> 00:22:48,310
of 1 minus y squared.

384
00:22:48,310 --> 00:22:52,540

385
00:22:52,540 --> 00:22:55,350
That's sort of unexpected.

386
00:22:55,350 --> 00:22:59,000
This function has
this derivative.

387
00:22:59,000 --> 00:23:02,440
This function has the
same derivative but

388
00:23:02,440 --> 00:23:03,690
with a minus sign.

389
00:23:03,690 --> 00:23:06,810

390
00:23:06,810 --> 00:23:10,740
That suggests that somehow if I
add those, yeah, let's just

391
00:23:10,740 --> 00:23:13,750
think about that for the
last minute here.

392
00:23:13,750 --> 00:23:21,480
That says that if I add sine
inverse y to cosine inverse y,

393
00:23:21,480 --> 00:23:25,070
their derivatives will cancel.

394
00:23:25,070 --> 00:23:29,100
So the derivative of that
sum of this one--

395
00:23:29,100 --> 00:23:32,100
can I do a giant plus
sign there?--

396
00:23:32,100 --> 00:23:33,490
is 0.

397
00:23:33,490 --> 00:23:36,580
The derivative of that plus the
derivative of that is a

398
00:23:36,580 --> 00:23:40,350
plus thing and a minus
thing, giving 0.

399
00:23:40,350 --> 00:23:41,600
So how could that be?

400
00:23:41,600 --> 00:23:45,500

401
00:23:45,500 --> 00:23:48,580
Have you ever thought
about what functions

402
00:23:48,580 --> 00:23:52,560
have derivative 0?

403
00:23:52,560 --> 00:23:55,120
Well, actually, you have.
You know what

404
00:23:55,120 --> 00:23:57,890
functions have no slope.

405
00:23:57,890 --> 00:23:59,670
Constant functions.

406
00:23:59,670 --> 00:24:05,270
So I'm saying that it must
happen that the arc sine

407
00:24:05,270 --> 00:24:11,540
function plus the arc cosine
function is a constant.

408
00:24:11,540 --> 00:24:14,490
Then its derivative
is 0, and we are

409
00:24:14,490 --> 00:24:16,510
happy with our formulas.

410
00:24:16,510 --> 00:24:18,770
And actually, that's true.

411
00:24:18,770 --> 00:24:24,410
The arc sine function
gives me this angle.

412
00:24:24,410 --> 00:24:28,320
The arc cosine function
would give me--

413
00:24:28,320 --> 00:24:32,580
shall I give that angle another
name like alpha?

414
00:24:32,580 --> 00:24:35,750
This one would be the theta.

415
00:24:35,750 --> 00:24:38,260
That one would be the alpha.

416
00:24:38,260 --> 00:24:43,690
And do you believe that in that
triangle theta plus alpha

417
00:24:43,690 --> 00:24:47,320
is a constant and therefore
has derivative 0?

418
00:24:47,320 --> 00:24:50,560
In fact, yes, you
know what it is.

419
00:24:50,560 --> 00:24:54,870
Theta plus alpha in a right
triangle, if I add that angle

420
00:24:54,870 --> 00:24:58,600
and that angle, I
get 90 degrees.

421
00:24:58,600 --> 00:25:01,350
A constant.

422
00:25:01,350 --> 00:25:03,840
Well, 90 degrees, but I
shouldn't allow myself to

423
00:25:03,840 --> 00:25:04,880
write that.

424
00:25:04,880 --> 00:25:06,920
I must write it in radians.

425
00:25:06,920 --> 00:25:08,170
A constant.

426
00:25:08,170 --> 00:25:13,490

427
00:25:13,490 --> 00:25:23,400
OK, don't forget the great
result from today.

428
00:25:23,400 --> 00:25:29,220
We filled in the one power that
was missing, and we're

429
00:25:29,220 --> 00:25:30,220
ready to go.

430
00:25:30,220 --> 00:25:32,740
Thank you.

431
00:25:32,740 --> 00:25:34,550
NARRATOR: This has been
a production of MIT

432
00:25:34,550 --> 00:25:36,940
OpenCourseWare and
Gilbert Strang.

433
00:25:36,940 --> 00:25:39,210
Funding for this video was
provided by the Lord

434
00:25:39,210 --> 00:25:40,430
Foundation.

435
00:25:40,430 --> 00:25:43,560
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436
00:25:43,560 --> 00:25:46,640
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437
00:25:46,640 --> 00:25:48,200
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438
00:25:48,200 --> 00:25:50,326