PROFESSOR: OK, earlier lecture
introduced the logarithm as
the inverse function
to the exponential.
And now it's time to do
calculus, find its derivative.
And we spoke about other inverse
functions, and here is
an important one: the inverse
sine function, or sometimes
called the arc sine.
We'll find its derivative,
too.
OK, so what we're doing really
is sort of completing the list
of important rules
for derivatives.
We know about the derivative
of a sum.
Just add the derivatives.
f minus g, just subtract
the derivatives.
We know the product rule and the
quotient rule, so that's
add, subtract, multiply,
divide functions.
And then very, very important is
the chain of functions, the
chain rule.
This is never to be mixed
up with that.
You wouldn't do such a thing.
And now we're adding
one more f inverse.
That's today, the derivative
of the inverse.
That will really complete
the rules.
Out of simple functions like
exponential, sine and cosine,
powers of x, this creates all
the rest of the functions that
we typically use.
OK, so let's start with the most
important of all: the f
of x is e to the x.
And then the inverse
function, we
named the natural logarithm.
And notice, remember how
I reversed the letters.
Here, x is the input and y
is the output from the
exponential function.
So for the inverse function,
y is the input.
I go backwards to x, and the
thing to remember, the one
thing to remember, just tell
yourself x is the exponent.
The logarithm is the exponent.
OK, so a chain of functions
is coming here, and it's a
perfectly terrific chain.
This is the rule for
inverse functions.
If I start with an x and I do
f of x, that gets me to y.
And now I do the inverse
function, and it
brings me back to x.
So I really have a chain
of functions.
The chain has a very
special result.
And our situation is if we
know how to take the
derivative of f, this ought to
tell us-- the chain rule--
how to take the derivative
of the
inverse function, f inverse.
Let me try it with this
all-important example.
So which--
and notice also, chain
goes the other way.
If I start with y, do the
inverse function, then
I've reached x.
Then f of x is y.
Maybe before I take e to the
x, let me take the function
that always is the
starting point.
So practice is--
I'll call this example of--
just to remember how inverse
functions work--
linear functions.
y equals ax plus b.
That's f of x there.
Linear.
What's the inverse of that?
Now the point about inverses
is I want to solve for x in
terms of y.
I want to get x by itself.
So I move b to the
opposite side.
So in the end, I want to
get x equals something.
And how do I do that?
I move b to the opposite side,
and then I still have ax, so I
divide by a.
Then I've got x by itself.
This is the inverse.
This is f inverse of y.
Notice something about inverse
functions, that here we did--
this function f of x was
created in two steps.
it was sort of a chain
in itself.
The first step was
multiply by a.
We multiply and then we add.
What's inverse function do?
The inverse function takes y.
Subtracts b.
So it does subtract first
to get y minus
b, and then it divides.
What's my point?
My point is that if a function
is built in two steps,
multiply and then add in this
nice case, the inverse
function does the
inverse steps.
Instead of multiplying
it divides.
Instead of add, it subtracts.
What it does though is
the opposite order.
Notice the multiply was done
first, then the divide is
last. The add was done second,
the subtract was done first.
When you invert things, the
order-- well, you know it.
It has to be that way.
It's that way in life, right?
If we're standing on the beach
and we walk to the water and
then we swim to the dock, so
that's our function f of x
from where we were to the dock,
then how do we get back?
Well, wise to swim
first, right?
You don't want to walk first.
From the dock, you swim back.
So you swam one way at the
end, and then in the
inverse, you swim--
oh, you get it.
OK, so now I'm ready
for the real thing.
I'll take one of those chains
and take its derivative.
Let me take the first one.
So the first one says that the
log of e to the x is x.
That's--
The logarithm was defined
that way.
That's what the log is.
Let's take the derivative of
that equation, the derivative
of both sides.
We will be learning what's the
derivative of the log.
That's what we don't know.
So if I take the derivative,
well, on the right-hand side,
I certainly get 1.
On the left-hand side, this
is where it's interesting.
So it's the chain rule.
The log of something, so I
should take the derivative--
oh, I need a little more
space here, over here.
The derivative of log
y, y is e to the x.
Everybody's got that, right?
So this is log y, and I'm taking
its derivative with
respect to y.
But then I have to, as the chain
rule tells me, take the
derivative of what's inside.
What's inside is e to the
x, so I have dy dx.
So I put dy dx.
And the derivative on the
right-hand side, the neat
point here is that the
x-derivative of that is 1.
OK, now I'm going to learn what
this is because I know
what this is: the derivative
of dy dx, the derivative of
the exponential.
Well, now comes the most
important property that we use
to construct this exponential.
dy dx is e to the x.
No problem, OK?
And now, I'm going to divide
by it to get what I want--
almost. Almost, I say.
Well, I've got the derivative
of log y here.
Correct, but there's a
step to take still.
I have to write--
I want a function of y.
The log is a function of y.
It's derivative is--
the answer is a function of y.
So I have to go back
from x to y.
But that's simple.
e to the x is y.
Oh!
Look at this fantastic answer.
The derivative of the log, the
thing we wanted, is 1 over y.
Why do I say fantastic?
Because out of the blue almost,
we've discovered the
function log y, which has that
derivative 1 over y.
And the point is this is
the minus 1 power.
It's the only power that we
didn't produce earlier as a
derivative.
I have to make that point.
You remember the very first
derivatives we knew were the
derivatives of x to the
n-th, powers of x.
Everybody knows that that's n
times x to the n minus 1.
The derivative of every power
is one power below.
With one exception.
With one exception.
If n is 0, so I have to
put except n equals 0.
Well, it's true when n is 0,
so I don't mean the formula
doesn't fail.
What fails is when n is 0, this
right-hand side is 0, and
I don't get the minus 1 power.
No power of x produces the minus
1 power when I take the
derivative.
So that was like an open hole
in the list of derivatives.
Nobody was giving the derivative
to be the minus 1
power when we were looking
at the powers of x.
Well, here it showed up.
Now, you'll say the
letter y's there.
OK, that's the 25th letter
of the alphabet.
I'm perfectly happy if you
prefer the 26th letter.
You can write d log z dz equals
1/z if you want to.
You can write, as you might
like to, d by dx.
Use the 24th letter
of log x is 1/x.
I'm perfectly OK for you to do
that, to write the x there,
now after we got the formula.
Up to this point, I really had
to keep x and y straight
because I was beginning
from y is e to the x.
That was my starting point.
OK, so that keeping them
straight got me the derivative
of log y as 1/y.
End.
Now, I'm totally happy if you
use any other letter.
Use t if you have
things growing.
And remember about the
logarithm now.
We can see why it
grows so slowly.
Because its slope is 1/y.
Or let's look at this one,
because we're used to thinking
of graphs with x
along the axis.
And this is telling us that the
slope of the log curve--
the log curve is increasing, but
the slope is decreasing,
getting smaller and smaller.
As x gets very small, it's
just barely increasing.
It does keep going on
up to infinity,
but very, very slowly.
And why is that?
That's because the exponential
is going very, very quickly.
And you remember that the one
graph is just the flip of the
other graph, so if one is
climbing like mad, the other
one is growing slowly.
OK, that's the main facts, the
most important formula of
today's lecture.
I could--
do you feel like practice to
take the chain in the opposite
direction just to see
what would happen?
So what's the opposite
direction?
I guess the opposite direction
is to start with--
which did I start with?
I started with log of
e to the x is x.
The opposite direction would be
to start with e to the log
y is y, right?
That's the same chain.
That's the f inverse coming
before the f.
What do I do?
Take derivatives.
Take the derivative
of everything, OK?
So take the derivative,
the y-derivative.
I get the nice thing.
I mean, that's the fun part,
taking the derivative on the
right-hand side.
On the left side, a little more
work, but I know how to
take the derivative of
e to the something.
It's the chain rule.
Of course it's the chain rule.
We got a chain here.
So the derivative of e to the
something, now you remember
with the chain rule, is e to
that same something times the
derivative of what's inside.
The derivative and what's
inside is this guy: the
derivative of log y dy.
This is what we want to know,
the one we know, and what is e
to the log y?
It's sitting up there
on the line before.
e to the log y is y.
So this parenthesis is
just containing y.
Bring it down.
Set it under there, and
you have it again.
The derivative of log y
dy is 1 over e to the
log y, which is y.
OK, we sort of have done more
about inverse functions than
typical lectures might, but I
did it really because they're
kind of not so simple.
And yet, they're crucially
important in this situation of
connecting exponential
with log.
And by the way, I prefer to
start with exponential.
The logic goes also just fine.
In fact, some steps are a little
smoother if you start
with a logarithm function,
define that somehow, and then
take its inverse, which will
be the exponential.
But for me, the exponential
is so all important.
The logarithm is important, but
it's not in the league of
e to the x.
So I prefer to do it this
way to know e to the x.
Now if you bear with me,
I'll do the other
derivative for today.
The other derivative
is this one.
Can we do that?
OK, so I want the derivative
of this arc sine
function, all right?
So I'm going to--
let me bring that.
This side of the board is now
going to be x is the inverse
sine of y, or it's often called
the arc sine of y.
OK, good.
All right.
So again, I have a chain.
I start with x.
I create y.
So y is sine x.
So y is the sine of x, but
x is the arc sine of y.
That's the chain.
Start with a y.
Do f inverse.
Do f, and you got y
again, all right?
Now, I'm interested in the
derivative, the derivative of
this arc sine of y.
I want the y-derivative.
I'm just going to copy this
plan, but instead of e, I've
got sines here.
So take the y-derivative of both
sides, the y-derivative
of both sides.
Well, I always like that one.
The y-derivative of this
is the chain rule.
So I have the sine of some
inside function.
So the derivative is the cosine
of that inside function
times the derivative of the
inside function, which is the
guy we want.
OK, so I have to figure
out that thing.
In other words, I guess I've
got to think a little bit
about these inverse
trig functions.
OK, so what's the story with
the inverse trig functions?
The point will be this
is an angle.
Ha!
That's an angle.
Let me draw the triangle.
Here is my angle theta.
Here is my sine theta.
Here is my cos theta, and
everybody knows that now the
hypotenuse is 1.
So here is theta.
OK, whoa!
Wait a minute.
I would love theta to
be the angle whose--
oh, maybe it is.
This is the angle whose sine--
theta should be the angle
whose sine is y, right?
OK, theta is the angle
whose sine is y.
OK, let me make that happen.
And now, tell me the other side
because I got to get a
cosine in here somewhere.
What is this side?
Back to Pythagoras, the
most important fact
about a right triangle.
This side will be the
square root of--
this squared plus this squared
is 1, so this is the square
root of 1 minus y squared.
And that's the cosine.
The cosine of this angle theta
is this guy divided by 1.
We're there, and all I've used
pretty quickly was I popped up
a triangle there.
I named an angle theta.
I took its sine to be y, and
I figured out what its
cosine had to be.
OK, so there's the theta.
Its cosine has to be this,
and now I'm ready to
write out the answer.
I'm ready to write down
the answer there.
That has a 1 equals--
the cosine of theta, that's this
times the derivative of
the inverse sine.
You see, I had to get this
expression into something--
I had to solve it for y.
I had to figure out what
that quantity is as
a function of y.
And now I just put
this down below.
So if I cross this out
and put it down here,
I've got the answer.
There is the derivative of the
arc sine function: 1 over the
square root of 1 minus
y squared.
OK, it's not as beautiful as
1/y, but it shows up in a lot
of problems. As we said earlier,
sines and cosines are
involved with repeated motion,
going around a circle, going
up and down, going across
and back, in and out.
And it will turn out that this
quantity, which is really
coming from the Pythagoras, is
going to turn up, and we'll
need to know that it's the
derivative of the arc sine.
And may I just write down what's
the derivative of the
arc cosine as long
as we're at it?
And then I'm done.
The derivative of the
arc cosine, well,
you remember what--
what's the difference between
sines and cosines when we take
derivatives?
The cosine has a minus.
So there'll be a minus 1
over the square root
of 1 minus y squared.
That's sort of unexpected.
This function has
this derivative.
This function has the
same derivative but
with a minus sign.
That suggests that somehow if I
add those, yeah, let's just
think about that for the
last minute here.
That says that if I add sine
inverse y to cosine inverse y,
their derivatives will cancel.
So the derivative of that
sum of this one--
can I do a giant plus
sign there?--
is 0.
The derivative of that plus the
derivative of that is a
plus thing and a minus
thing, giving 0.
So how could that be?
Have you ever thought
about what functions
have derivative 0?
Well, actually, you have.
You know what
functions have no slope.
Constant functions.
So I'm saying that it must
happen that the arc sine
function plus the arc cosine
function is a constant.
Then its derivative
is 0, and we are
happy with our formulas.
And actually, that's true.
The arc sine function
gives me this angle.
The arc cosine function
would give me--
shall I give that angle another
name like alpha?
This one would be the theta.
That one would be the alpha.
And do you believe that in that
triangle theta plus alpha
is a constant and therefore
has derivative 0?
In fact, yes, you
know what it is.
Theta plus alpha in a right
triangle, if I add that angle
and that angle, I
get 90 degrees.
A constant.
Well, 90 degrees, but I
shouldn't allow myself to
write that.
I must write it in radians.
A constant.
OK, don't forget the great
result from today.
We filled in the one power that
was missing, and we're
ready to go.
Thank you.
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