Partial Derivatives

Sorry to bother again. http://i34.tinypic.com/17wq2u.jpg

Either I'm missing something about dealing with these expressions or this makes no sense whatsoever. I can upload the rest of the question if necessary; it's about transforming the expression "∂2f/∂x2 + ∂2f/∂y2" to the variables g, ρ and Φ where:

f(x,y) is the function in the Cartesian plane
g(ρ,Φ) gives the function in polar coordinates
and
x = ρ cos Φ
y = ρ sin Φ

But the thing in the pic is the only part I don't get.

The d/dΦ is an operator.

You can multiply out the brackets but remember when you "multiply" the sinΦ/p.d/dΦ term by the cos Φ . dg/dρ term the operator will kind of produce two terms because of the product rule,

so you get (sinΦ/p.d/dΦ)(cos Φ . dg/dρ)

= sin Φ cos Φ dg/dρdΦ - sin ^2/p dg / dρ


So for example the 1/ρ^2 term is coming in because the operator d/dρ is going to act on the term 1/ρ as well as the term dg/dΦ

That make it clear for you?