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\[\frac{\partial^2f(x,y)}{\partial x \partial y}=\frac{\partial^2f(x,y)}{\partial y \partial x}\] Is this always true???
NOPPPPPe. It's true IF AND ONLY IF f(x,y) is a symmetric one.

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Other answers:

What's that mean ??
like \(f(x,y)= x^2 + y^2 + x +y\) what happens to x, it happens to y. if \(f(x,y) = x^2 + y \) then partial x, y is different from partial y, x
wait symmetric as in do u mean f(x,y)=f(y,x)??
kind of
Other example: \(f(x,y) = 2x^2 + 5xy + 2y^2\) partial w.r.t x = \(f'_x= 4x +5y\), then w.r.t y \(f"_{x,y}= 5\) partial w.r.t.y = \(f'_y =4y +5x\) then, w.r.t x \(f"_{y,x}= 5\)
to higher degree of a function, the letters switch to each other, but the answer are the same like \(f(x,y) = x^3+ 20x^2y + y^3\), it is a symmetric one, to find \(f"_{x,y}~~f"_{y,x}\), you can save time by doing just one, then switch the letters.
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hmm kk
according to the Gospel of Mary (Boas : Mathematical Methods in Physical Sciences [p190, 3rd Edition, if you have access] ), so long as \(f_x, f_y, f_{xx} and f_{yy}\) are continuous, then \(f_{xy} = f_{yx}\) here's an example of an exception: http://www.math.tamu.edu/~tvogel/gallery/node18.html and if you google something like "mixed partials not equal" i think you will find more stuff. my sense FWIW is that if you are doing physical/applied stuff you will be more than aware of the discontinuity and can probably even find a way around it.

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