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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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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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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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End.
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Now, I'm totally happy if you
use any other letter.
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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.
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Because its slope is 1/y.
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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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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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As x gets very small, it's
just barely increasing.
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It does keep going on
up to infinity,
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but very, very slowly.
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And why is that?
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That's because the exponential
is going very, very quickly.
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And you remember that the one
graph is just the flip of the
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other graph, so if one is
climbing like mad, the other
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one is growing slowly.
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OK, that's the main facts, the
most important formula of
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today's lecture.
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I could--
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do you feel like practice to
take the chain in the opposite
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direction just to see
what would happen?
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00:13:16,780 --> 00:13:20,560
So what's the opposite
direction?
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I guess the opposite direction
is to start with--
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which did I start with?
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I started with log of
e to the x is x.
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The opposite direction would be
to start with e to the log
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y is y, right?
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That's the same chain.
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That's the f inverse coming
before the f.
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What do I do?
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Take derivatives.
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Take the derivative
of everything, OK?
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So take the derivative,
the y-derivative.
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I get the nice thing.
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I mean, that's the fun part,
taking the derivative on the
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right-hand side.
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On the left side, a little more
work, but I know how to
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take the derivative of
e to the something.
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It's the chain rule.
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Of course it's the chain rule.
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We got a chain here.
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So the derivative of e to the
something, now you remember
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with the chain rule, is e to
that same something times the
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derivative of what's inside.
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The derivative and what's
inside is this guy: the
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derivative of log y dy.
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This is what we want to know,
the one we know, and what is e
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00:14:41,340 --> 00:14:42,590
to the log y?
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00:14:42,590 --> 00:14:45,430
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00:14:45,430 --> 00:14:48,190
It's sitting up there
on the line before.
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e to the log y is y.
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So this parenthesis is
just containing y.
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Bring it down.
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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
To help OCW continue to provide
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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