Can Symmetry Simplify Partial Differentiation for Multivariable Functions?

[Kolahal Bhattacharya]

given f=ln(x^3+y^3+z^3-3xyz)To prove df/dx+df/dy+df/dz=3/(x+y+z)
also finding (d^2/dx^2+...similar two more terms)f=? d => del
& (d^2/dx^2+...)^2f=?
I have done the first part of the problem.The trick is to write e^f=x^3+y^3+z^3-3xyz and then to differentiate.
However the next parts are coming hopelessly huge, by the same token.

Please help.

Kolahal Bhattacharya

 

[0rthodontist]

One thing that will speed it up a little is the symmetry between x, y, and z. Given the derivative for one, you can easily write down the derivative for the others just by swapping variables.