Can you explain the equality of mixed partial derivites? I can take partial derivitives, but what does show that Fxy = Fyx??

Question

anonymous · Answer

Verify that Fxy = Fyx

F(x, y) = cos(xy)

Fx = -ysin(xy)

Fy = -xsin(xy)

ganeshie8 · Answer

$\large  F_{xy}  =  (-y\sin(xy))_y $

use product rule

ganeshie8 · Answer

$F_{xy} =  -[\sin(xy) +  xy\cos(xy)]$

ganeshie8 · Answer

$F_{yx} =  (-x\sin(xy))_x$

take the partial with.respect.to x

kainui · Answer

So basically all you're doing is you're taking a function, taking the derivative with respect to x first, then y second. Then start over with the same function, except this time take the derivative with respect to y first, then x. tadah, they're the same.

ganeshie8 · Answer

he may be looking for proof or something ? @jasonjohnson86

usukidoll · Answer

Fxy means take the partial derivative of y first and then take the partial derivative of x (if any)

amistre64 · Answer

Fxy means (Fx)y ... partial wrt x first, then wrt y

it is only when placed in some fractional format that it is read right to left:$$\frac{\delta F}{\delta x \delta y}$$would be y first then x .... cant recall how to code up the funny 'd'

anonymous · Answer

Thanks so much for the help everyone I got it!