# Partial derivative helpWatch

#1
Does this:

∂^2z / ∂x∂y
(partial squared z / partial x partial y)

Mean the differentiate with respect to y first, then with respect to x or can I differentiate either variable first?

My function is: z = 4x^2 −xy +y^2 −x^3 and I get the answer as -1 whether I differentiate x or y first but that might be a coincidence.
0
1 week ago
#2
(Original post by HoldThisL)
Does this:

∂^2z / ∂x∂y
(partial squared z / partial x partial y)

Mean the differentiate with respect to y first, then with respect to x or can I differentiate either variable first?

My function is: z = 4x^2 −xy +y^2 −x^3 and I get the answer as -1 whether I differentiate x or y first but that might be a coincidence.
99% sure that it's with respect to x first and then with respect to y.

I also got them both the same no matter how I did it when calculating Hessian, but that again may just be a coincidence.
0
#3
(Original post by I'm God)
99% sure that it's with respect to x first and then with respect to y.

I also got them both the same no matter how I did it when calculating Hessian, but that again may just be a coincidence.
That would make sense but I found a University of Surrey resource (because my university doesn't teach jack) which gave the same equation but said differentiate y first. Perhaps it is interchangeable?
0
1 week ago
#4
(Original post by HoldThisL)
That would make sense but I found a University of Surrey resource (because my university doesn't teach jack) which gave the same equation but said differentiate y first. Perhaps it is interchangeable?
Okay, I changed your equation a bit and tried it with 4x^2y - xy + y^2 -x^3y. I did it both ways and got the same answer so it probably is interchangeable, but I would still take caution.
1
1 week ago
#5
(Original post by HoldThisL)
That would make sense but I found a University of Surrey resource (because my university doesn't teach jack) which gave the same equation but said differentiate y first. Perhaps it is interchangeable?
Actually, never mind, I just found a Khan Academy video called "Symmetry of second partial derivatives" which says that the order doesn't matter. You can check it out
0
#6
(Original post by I'm God)
Actually, never mind, I just found a Khan Academy video called "Symmetry of second partial derivatives" which says that the order doesn't matter. You can check it out
Cheers
0
1 week ago
#7
(Original post by HoldThisL)
That would make sense but I found a University of Surrey resource (because my university doesn't teach jack) which gave the same equation but said differentiate y first. Perhaps it is interchangeable?
(Original post by I'm God)
Okay, I changed your equation a bit and tried it with 4x^2y - xy + y^2 -x^3y. I did it both ways and got the same answer so it probably is interchangeable, but I would still take caution.
(Original post by I'm God)
Actually, never mind, I just found a Khan Academy video called "Symmetry of second partial derivatives" which says that the order doesn't matter. You can check it out
There's a standard result (Clairaut's theorem) that says that the order doesn't matter so long as f has continuous second partial derivatives.

This is nearly always true, but there are examples where this doesn't hold and the order does actually matter.

is an such an example, and you will find that at (0, 0)

, while

(See https://en.wikipedia.org/wiki/Symmet..._of_continuity )
Last edited by DFranklin; 1 week ago
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