Welcome to a video
on Second Order Partial Derivatives.
Just like we were able to determine higher order derivatives
of a function in one variable
we can also determine higher order partial derivatives
of functions in two or more variables.
And since there are two first order of partial derivatives
of functions of two variables
there are going to be four, second order partial derivatives
of functions of two variables.
In the notation used
for second order of partial derivatives
is very important.
Let's go ahead and take a look at that.
All the notation here is telling us to find the
partial derivative with respects to x twice.
Similarly, the notation here is telling us to find
the partial derivative with respects to y, two times.
Now these next two, can be a little tricky at first.
All the notation here is telling us to find
the partial derivative with respects to x
and then the partial derivative with respects to y.
In that order.
So, looking at these first two notations
we work our way from right to left.
We find the partial with respects to x
and then with respects to y
in both cases.
Now, looking at these last three notations
we find the partial with respects to x
and then with respects to y again.
We notice how the order of this notation here
is now from left to right.
And over here it's right to left.
So we differentiate with respects to x first
and then with respects to y.
And then here, the notation has reversed
to find the partial derivative with respects to y first
and then with respects x.
So we need to make sure that we are aware
of the differences here.
And sometimes these last two are called
the second order of mixed, partial derivatives.
Let's talk about the Geometric Interpretation
of the Second Order of Partial Derivatives.
Now sometimes we'll just say fxx and fyy
for these second order of partials.
If fxx is greater than zero
then the function is concave up in the x direction.
Which means if it's negative it'll be concave down
and if fyy is positive
then f of xy is concave up in the y direction.
And again, if it's negative, it'll be concave down.
In general, mixed partials tell us
how a partial in one variable
is changing in the direction of the other.
So if we have fxy, this tells us how the rate of change
of f of xy, in the x direction is changing
as we move in the y direction.
Let's take a look at a graphic that illustrates
these first two
referring to concavity.
Right now we have fxx and fyy are both positive.
Meaning, in the x direction
as we see here
the surface is concave up
as well as in the y direction, as we see here.
Now let's go ahead and change it
so that, fxx is negative
and fyy remains positive.
So in the x direction, you can see now it's concave down
in the y direction it's still
concave up.
Now let's go ahead and make fxx
positive
and fyy negative.
So now in the x direction you can see it's concave up
and the y direction, it is concave down.
Let's go ahead and take a look at some examples
to get some practice on finding
second order partial derivatives.
Before we do that, we have to find the
first order of partial derivatives.
So the first partial with respects to x
will treat y as a constant, we'll have
eight x to the third
minus 15, x squared, y to the third.
Which should be zero.
First partial with respects to y we'd have zero here
negative 15, x cubed, y squared.
Minus four y cubed.
Now let's go ahead and find all four
second order of partial derivatives.
So here we're going to take this first partial derivative
and differentiate it again with respects to x.
So we'll have 24 x squared
minus 30 x to the first, y to the third.
For fyy, we're going to find the derivative of this
with respects to y again.
So we're going to have negative 30, x cubed, y to the first.
Minus, 12 y squared.
Now for this mixed partial, we'll find
the partial derivative with respects to x
and then with respects to y.
So here's the partial with respects to x
and now we'll differentiate this with respects to y
treating x as a constant.
So this would be zero
and this would be, multiply by three we're going to get
negative 45, x squared
and then y squared.
And now here, we'll differentiate
the first partial with respects to y
with respects to x.
So we'll multiply by three negative 45 x squared, y squared.
And here, we're differentiating with respects to x
so this would be zero.
One thing you'll notice is
if you take a look at the two mixed partials
they're equal to each other.
And in fact, they will always be equal to each other.
So let's go ahead and state that.
These two mixed partials, will always be equal
as long they aren't continuous
over a specific region.
So in the future, if we find one of these
we also know the other.
Let's take a look at one more example
that's a little bit more involved.
We'll start by determining our first
order of partial derivatives.
So the first partial with respects to x
is going to require the Product Rule
so we'll have x, times the derivative of, exy squared
so we'll have e xy squared
times u prime, treating y as a constant
we'll just get y squared.
Plus
the second
times derivative of the first, that would just be one.
Let's go ahead and clean this up.
Notice the exponential is a common factor
let's go ahead and factor that out.
So we'll be left with xy squared
and then plus one.
Let's go ahead and determine the first
partial with respects to y.
So we're treating x as a constant
so we'll just have xe to the xy squared
times u prime.
So derivative of the exponent with respects to y
would just be two xy.
Let's clean this up.
So we have two x squared, y e to the xy squared.
Now let's go ahead and find our second order of partials.
So for fxx, we'll differentiate this
with respects to x again, treating y as a constant.
So it's going to require the Product Rule.
Times derivative of the second
with respects to x, so we'll have
y squared, and then zero.
Plus, the second
times derivative of the first, again with respects to x.
So we'll have e xy squared.
The derivative of the exponent with respects to x
is going to be y squared.
Let's go ahead and clean this up.
We have to distribute here, and here.
Notice the second distribution, we're going to have another
of y squared, e to the xy squared
and we have another one here.
So we're actually going to have two of those.
And then we'll have x, y to the fourth
e to the xy squared.
To determine fyy
we're going to go ahead and find the
partial derivative of this, with respects to y again.
So notice we will have to apply the Product Rule again
so we'll have the first
times the derivative of the second
so we'll have e xy squared
times u prime.
So we'll have 2 xy
plus second
times derivative of the first, with respects to y
that'll give us two x squared.
So it looks like we're going to have four
x cubed, y squared.
E to the xy squared.
And then we'll have plus, two x squared
e to the xy squared.
I could factor this, but I'm going to go
and leave it like that.
Let's go and copy these first order of partials
on the next slide and then we'll find the
second order of mixed partials.
Remember what we discovered earlier
that these are going to be equal to each other
so we could really just find, one of them.
Let's go ahead and find
this mixed partial, and then we'll also know the other one.
So we're going to go ahead and take this first partial
with respects to x
and then differentiate again, with respects to y
treating x as a constant.
So [mumbles] the Product Rule
we'll have e xy squared
times the derivative of this, with respects to y
so we'll have two xy
plus, the second
times the derivative of the exponential.
So we'll have e xy squared, times
derivative of the exponent with respects to y
will be, two xy.
Here we have to distribute.
Looking at the second distribution
we'll have two xy, e to the xy
we have another one here.
So we'll have four xy, e to the xy squared.
Then we'll have plus two, x squared, y cubed
e to the xy squared.
And again, we know these are equal
so I'm going to go ahead and just state that these are equal.
And that's going to have to do it for this video.
I hope you found these examples helpful.