I Derivative of f() as a function of a Laplacian

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I need a little help with understanding a differential relationship between functions. If g and f are vector fields and f(g(x,y),q(x,y))=∇2g(x,y) How could you, if possible, express ∂f/∂g explicitly? Please help a bit confused.
 

Delta2

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I have an idea, to take the derivative of f with respect to x (or y) and use the chain rule.

The answer will give ##\frac{\partial f}{\partial g}## in terms of partial derivatives of g ( ##\frac{\partial g^3}{\partial x\partial y^2},\frac{\partial g^3}{\partial x^3},\frac{\partial g}{\partial x}##), partial derivatives of q ##\frac{\partial q}{\partial x}##, AND ##\frac{\partial f}{\partial q}##. So we need to have some info about function q, and also know ##\frac{\partial f}{\partial q}##.
 

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