Laplace operator

Question

The question is that

Derive a formula for $\Delta(\frac{f}{g})$ in terms of $f, g, \nabla f, \nabla g, \Delta f, \Delta g$.

Naturally, I apply the rules of gradient and divergence, and yield $$\Delta(\frac{f}{g}) = \frac{\Delta f}{g} + 2 \nabla f \cdot \nabla (\frac{1}{g}) + f\Delta (\frac{1}{g})$$ However this is the very far I can get. This result seems not answering the question properly and is not satisfactory to further questions. Please help and thanks in advance.

Answer 1

Hint: you should start from the quotient rules

$$\Delta(\frac{f}{g}) =\nabla \cdot (\nabla \frac{f}{g})=\nabla \cdot (\frac{g\nabla f-f\nabla g}{g^2})=\frac{\nabla\cdot(g\nabla f-f\nabla g)g^2-(g\nabla f-f\nabla g)\cdot \nabla(g^2) }{g^4}$$