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

[dsaun777]

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]

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 $\frac{\partial f}{\partial q}$.