Partial Derivatives Question

Hi there,

This is from an econ textbook, but the content is all math. The example in the book starts from the following identity:

D(p) = S(p,a)

Where p and a are variables. Furthermore p = p(a), so the new equation given is:

D(p(a)) = S(p(a), a).

The following part is where the book loses me, and they do not do a very good job of showing how these equations are derived. I haven't been exposed to partial derivatives yet, but know what they are. I just don't understand how the book gets this derivative:

(dD(p(a))/dp) dp/da = (partialdS(p(a), a))/(partialdp) (dp/da) + (partialdS(p(a), a))/(partialda)

Sorry if that is really messy, but I'm not sure how to use the tags. This isn't a homework problem or anything, I just really want to know how that equation is derived. Any help would be greatly appreciated.

I think I was just confused by the notation, but if anybody was going to post in this thread, let me know if this is right:

The right-hand side is the two partial derivatives added together while the left-hand side is just a regular derivative? The intention is to set the demand function (left) equal to the supply function (right) to achieve equilibrium. Since the supply function is a function of two variables, the partials need to be taken. So the right-hand side is the sum of those partial derivatives? Is this correct?