Coordinate Transform partials question

I wish to go from cartesian to cylindrical coordinates using the chain rule. I see here that

$x = rcos(\phi)$
$y = r sin(\phi)$

$r = \sqrt{x^2 + y^2}$
$\phi = arctan(\frac{y}{x})$

I am having difficulties interpreting this differential

$$\frac{ \partial}{\partial x} = \frac{ \partial r}{\partial x}\frac{ \partial}{\partial r} + \frac{ \partial \phi}{\partial x}\frac{ \partial}{\partial \phi}$$

Intuitively this makes sense as x is a function of $\rho$ and $\phi$ only. What I don't understand is where exactly the two terms on the right came from. Was it some intermediate operation that was performed? There must be some form of the product rule at work, but I do not see it. Thanks

-
check this out: en.wikipedia.org/wiki/Jacobian_matrix_and_determinant –  Lucas Dec 14 '12 at 4:32