When computing the matrices of partial derivatives, does the row and the column of the matrix matter? For example if you are asked to compute the partial derivative of the following matrix
f(x,y) = ( x+y, x-y, xy) and f(x,y,z) = (x+z, y-5z, x-y), how will you arrange the matrix? My problem is not computing the partial derivative but how the partial derivative matrices should be arranged. Thanks
f(x,y) = ( x+y, x-y, xy) and f(x,y,z) = (x+z, y-5z, x-y), how will you arrange the matrix? My problem is not computing the partial derivative but how the partial derivative matrices should be arranged. Thanks
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Each row of the derivative matrix comes from all possible first partial derivatives of a given entry of f.
For instance, Df equals
[(∂/∂x)(x+y)...(∂/∂y)(x+y)...(∂/∂z)(x+…
[(∂/∂x)(x-y)....(∂/∂y)(x-y)....(∂/∂z)(…
[(∂/∂x)(xy).....(∂/∂y)(xy).....(∂/∂z)(…
This simplifies to
[1 1 0]
[1 -1 0]
[y x 0].
Similarly, for the second one, you should get
[1 0 1]
[0 1 -5]
[1 -1 0].
I hope this helps!
For instance, Df equals
[(∂/∂x)(x+y)...(∂/∂y)(x+y)...(∂/∂z)(x+…
[(∂/∂x)(x-y)....(∂/∂y)(x-y)....(∂/∂z)(…
[(∂/∂x)(xy).....(∂/∂y)(xy).....(∂/∂z)(…
This simplifies to
[1 1 0]
[1 -1 0]
[y x 0].
Similarly, for the second one, you should get
[1 0 1]
[0 1 -5]
[1 -1 0].
I hope this helps!