chain rule

jsbeckton

Junior Member
Joined
Oct 24, 2005
Messages
174
have a question about a chain rule problem thats got me stuck:

let f be a real valued differentiable function. Show that

Code:
z = f (x^2+y^2) + xy  satisfies y(dz/dx)-x(dz/dy) = y^2 - x ^2

how do i use the chain rule without knowing what x and y are? or am I supposed to use other varaibles such as s=x^s and t = y^2 ?
 
z=f[x^2+y^2]+xy take derivative with respect to x

1) dz/dx= f'[x^2+y^2] 2x +y
take derivative with respect to y
2) dz/dy = f ' [x^2+y^2] 2y +x

does y dz/dx -xdz/dy = ? y^2-x^2
y[f ' [x^2+y^2]2x+y] -x[f ' [x^2+y^2] 2y+x] = ?y^2-x^2
2xy f '[x^2+y^2] +y^2 -2xy f ' [x^2+y^2] -x^2 = ?[y^2-x^2
y^2-x^2= y^2-x^2 yes

another way to get at dz/dx or dz/dy
dz =[ f ' [x^2+y^2]2x + y] dx +{ f ' [x^2+y^2] 2y +x}dy

Arthur
 
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