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Total Differentiation

So the question is to find the stationary points and distinguish what type they are. The first one is:

z = (x^2)(y^2)-(x^2)-(y^2)

I understand that we must first partially differentiate in respect to both x and y, and so I did, getting:

dz/dz(Can't find partial sign) = (2*x*(y^2))-(2*x)
dz/dy = (2*y*(x^2))-(2*y)

And then I factorise both by making them each equal zero and get x=0,1,-1 and y=0,1,-1.

So my question is how do I work out the total derivative (i.e ((d^2)*z)/(dz*dy))?

I understand the equation is dz = fx*dx + fy*dy and I now know the anwser is 4*x*y, but do not know how to get there. Please could you go through it step by step if possible!

Thanks in advance.
Reply 1
Avatar for davros
davros
16
Original post by EconWill
So the question is to find the stationary points and distinguish what type they are. The first one is:

z = (x^2)(y^2)-(x^2)-(y^2)

I understand that we must first partially differentiate in respect to both x and y, and so I did, getting:

dz/dz(Can't find partial sign) = (2*x*(y^2))-(2*x)
dz/dy = (2*y*(x^2))-(2*y)

And then I factorise both by making them each equal zero and get x=0,1,-1 and y=0,1,-1.

So my question is how do I work out the total derivative (i.e ((d^2)*z)/(dz*dy))?

I understand the equation is dz = fx*dx + fy*dy and I now know the anwser is 4*x*y, but do not know how to get there. Please could you go through it step by step if possible!

Thanks in advance.


Assume my d's are partial too!

If you have dz/dx then just differentiate w.r.t.y to get d^2z/dxdy
Similarly, if you have dz/dy, diff w.r.t.x to get d^2z/dydx.

For well-behaved functions, these should both come out with the same answer!
Reply 2
Avatar for Ateo
Ateo
9
I don't see why you're trying to find the total derivative. You should type all of this in LaTeX, its not quite clear.

I presume that you intend to find ∂2z∂x∂y\displaystyle \frac{\partial ^2 z}{\partial x \partial y} . Note that this is just ∂∂x(∂z∂y)=4xy\displaystyle \frac{\partial }{\partial x} (\frac{\partial z}{\partial y}) = 4xy .
Reply 3
Avatar for ztibor
ztibor
10
Original post by EconWill
So the question is to find the stationary points and distinguish what type they are. The first one is:

z = (x^2)(y^2)-(x^2)-(y^2)

I understand that we must first partially differentiate in respect to both x and y, and so I did, getting:

dz/dz(Can't find partial sign) = (2*x*(y^2))-(2*x)
dz/dy = (2*y*(x^2))-(2*y)

And then I factorise both by making them each equal zero and get x=0,1,-1 and y=0,1,-1.

So my question is how do I work out the total derivative (i.e ((d^2)*z)/(dz*dy))?

I understand the equation is dz = fx*dx + fy*dy and I now know the anwser is 4*x*y, but do not know how to get there. Please could you go through it step by step if possible!

Thanks in advance.



You need for second partial derivatives of
∂2z∂x2\displaystyle \frac{\partial^2 z}{\partial x^2}
∂2z∂y2\displaystyle \frac{\partial^2 z}{\partial y^2}
∂2z∂x⋅∂y\displaystyle \frac{\partial^2 z}{\partial x \cdot \partial y}

Substituting the (x,y) coordinates in the second derivatives you can write the Hessian matrix (H)
(5 times for the 5 possible points (x,y) where the first derivatives were zero)

THe Det(H) and the sign of z"xxz"_{xx} show wthether there is stationary point in the relating (x,y), and if so, then which type is that.
(edited 10 years ago)

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