# If f is a differentiable function of three variables. Suppose w=f(x-y, y-z, z-x). Show that (dw/dx) + (dw/dy) + (dw/dz)=0

sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to find partial derivatives of function `w=f(x-y, y-z, z-x)`  such that:

`(del w)/(del x) = (del w)/(del f)*(del f)/(del x)`

`(del w)/(del x) = (del w)/(del f)*(1-1)`

`(del w)/(del x) = (del w)/(del f)*0`

`(del w)/(del x) = 0`

`(del w)/(del y) = (del w)/(del f)*(del f)/(del y)`

`(del w)/(del y) = (del w)/(del f)*(-1+1)`

`(del w)/(del y) = (del w)/(del f)*(0)`

`(del w)/(del y) = 0`

`(del w)/(del z) = (del w)/(del f)*(del f)/(del z)`

`(del w)/(del z) = (del w)/(del f)*(-1+1)`

`(del w)/(del z) = (del w)/(del f)*(0)`

`(del w)/(del z) = 0`

You need to add partial derivatives such that:

`(del w)/(del x) + (del w)/(del y) + (del w)/(del z) = 0+0+0= 0`

Hence, evaluating the partial derivatives of function  `w=f(x-y, y-z, z-x)`  yields that `(del w)/(del x) + (del w)/(del y) + (del w)/(del z) = 0.`