Partial Derivatives Help

Hi, welcome to the forum.

I'll use d for the symbol of the partial derivative.

d²w / dxdy = d/dx(dw/dy) = d/du(dw/dy)*du/dx + d/dv(dw/dy)*dv/dx = ...

(since du/dx = dv/dx = 1)

... = d/du(dw/du*du/dx + dw/dv*dv/dx) + d/dv(dw/du*du/dy + dw/dv*dv/dy) = ...

(since du/dy = 1 and dv/dy = -1)

...= d/du(dw/du + dw/dv) + d/dv(dw/du - dw/dv) = (d²w/du² + d²w/dudv) + (d²w/dvdu - d²w/dv²) = ...

(since d²w/dudv = d²w/dvdu)

... = d²w/du² - d²w/dv² = = d²f/du² - d²f/dv²