Hi, everyone! Welcome back to integralcalc.com.
Today we’re going to be talking about how
to calculate second order partial derivatives.
And in this particular problem, we’ve been
given the function f of x,y equals x squared
plus e to the y squared.
So notice first of all, that we’ve got a
function in two variables. We have both the
x variable and the y variable involved in
our function. And remember that if we’re
going calculate the second order partial derivatives,
we’ll need to calculate the second order
partial derivative with respect to x, with
respect to y and the mixed second order partial
derivative. Now, in order to calculate all
three of those second order partial derivatives,
we’ll first need to calculate the first
order partial derivatives because the second
order derivatives build off of the first order
derivatives. So for this video, I’m going
to assume that you have at least a basic understanding
of partial derivatives and that you know how
to calculate the first order derivatives but
of course, I’ll go through these steps as
a refresher.
So the first order partial derivatives are
going to be partial derivatives of f with
respect to x and with respect to y. So we’ll
calculate the partial derivative with respect
to x first. Remember that when you’re calculating
the partial derivative with respect to x,
you’re treating x as the variable and you’re
holding y constant. You’re treating it as
a constant, like it’s a number like 2 or
4 or something like that. So when we take
the partial derivative with respect to x,
we’ll go term by term. So taking the partial
derivative with respect to x of x squared
gives us 2x. Nothing fancy there. With respect
to e to the y squared, there’s no x variable
involved in that term. We’re treating y
as a constant so e y squared just becomes
zero because there’s no variable involved
there. So this is no different than saying
plus 4. We treat y as a constant. This is
a constant number here and so the derivative
is zero. So that will go away there. The partial
derivative with respect to y again term by
term. For the first term here, x squared,
there’s no y variable involved in that term.
So the derivative there is going to be zero.
But with respect to e to the y squared, we’ll
have to take the derivative with respect to
y. So remember that when we’re taking the
derivative of e to the x, the derivative is
just e to the x. So similarly, the derivative
will just be e to the y squared. But we need
to apply chain rule and multiply by the derivative
of the exponent, y squared. So the derivative
of y squared is just 2y. So we go ahead and
multiply by 2y. So now we see that these first
order derivatives are equal to 2x and the
one with respect to y is equal to 2y e to
the y squared.
Alright. So we’ve got out first order partial
derivatives and now we need to go ahead and
calculate our second order partial derivatives.
So we’ll go ahead and over here write the
second order partial derivative with respect
to x and then with respect to y. So in calculating
second order partial derivatives, all we’re
doing is taking the derivative of the first
order partial derivative. In this case, because
we took the first order partial derivative
with respect to x, here, we’re going to
take the partial derivative with respect to
x again. So we’re going to take this value
here and take the derivative with respect
to x again. When we do that, the derivative
of 2x is just 2. So the second order partial
derivative with respect to x is 2. And then
same thing here with y. We took the first
order partial derivative with respect to y
and we’ve got 2y e to the y squared. To
calculate the second order partial derivative
with respect to y, we’re just going to take
the derivative with respect to y again. This
time, we have to use product rule where we
have one function equal to 2y and the other
equal to e to the y squared. So remember that
product rule tells you to take the derivative
of one function multiply it by the other and
then add to that the derivative of the opposite
function multiplied by the original opposite
function. So what we’re going to do is take
the derivative first of 2y which is going
to be 2 and then we’ll multiply that by
e to the y squared. So we took the derivative
of this one leaving the second one alone and
then add to that this time, we’ll leave
2y alone but multiply by the derivative of
e to the y squared. Well we already know from
the original problem that that’s 2y e to
the y squared. So when we simplify this, we’ll
get 2e to the y squared plus 4y squared e
to the y squared and if we wanted to, we could
go ahead and factor out a 2e to the y squared
but we don’t have to. We would get 1 plus
2y squared. So either way, both of these are
correct. Either one, factored or not.
So we now have our second order partial derivatives
with respect to x and with respect to y. And
now all we need to do is calculate the mixed
second order partial derivative. And what
that looks like in terms of notation is second
order partial derivative of f, this time instead
of x squared or y squared, the notation looks
like this. Remember that before, we calculated
the first order partial derivative with respect
to x and then x again and here with respect
to y and then y again. When we do the mixed
derivative, we can do it either one of two
ways, we can take the first order partial
derivative with respect to x and calculate
the derivative of that with respect to y or
we can take the first order partial derivative
with respect to y and calculate the partial
derivative of that with respect to x. So we’re
just flipping the variable that we’re using.
It’s your choice and this is a good way
to check yourself because regardless of which
one you use, you should arrive at the same
answer for he mixed second order partial derivative.
If you don’t, you’ve probably done something
wrong because they always have to be the same.
So I would always recommend choosing the one
that looks easier. Obviously, this one here
looks easier to me. All we have to deal with
is 2x instead of 2y e to the y squared.
So let’s take the derivative of 2x with
respect to y. Well the derivative of 2x with
respect to y is just going to be zero because
there is no y variable involved in the function
2x. So the mixed second order partial derivative
is just zero. Notice that if we took the derivative
of this with respect to x, since there’s
no x variable involved in that function, we
would also get zero. So either way, this is
good news for us because we’ve confirmed
that these two match, which is what we want.
So we could have done it either way but we
chose 2x because it was simpler.
So again, when you’re talking about second
order partial derivatives, you want to calculate
all three – the second order partial derivative
with respect to x, with respect to y and the
mixed second order partial derivative. So
I hope that video helped you guys and I will
see you in the next one. Bye!
