Uploaded by TheIntegralCALC on 02.11.2011

Transcript:

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!

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!