This is exactly the rule I needed, Thank you so much!
So to start:
-∇⋅D(x,y)∇Φ(x,y) [3]
expands to
- (∇⋅D(x,y))⋅(∇Φ(x,y)) - D(x,y)ΔΦ(x,y) [4]
then
- (∂D/∂x+∂D/∂y)⋅(∂Φ/∂x+∂Φ/∂y) - D(x,y)(∂2Φ/∂x2+∂2Φ/∂y2)] [5]
distributing the lefthand term, removing null terms, and reducing to simpler...