You are making me feel old. My notation was orthodox in 1958. Indeed, in A Course Of Pure Mathematics, Tenth Edition, section 157, Hardy writes:

The distinction between the two functions is adequately shown by denoting the first by and the second by , in which case the theorem takes the form though this notation is also open to objection, in that it is a little misleading to denote the functions and whose forms as functions of x are quite different from one another, by the same letter f in and .

If q is a function of c, then h becomes a function of one independent variable, and your use of partials here doesn't make sense, because you can't hold c constant while changing q or vice versa.

I thought that this was the kind of situation partial derivatives are there for. AlanCrowe's just applied the multivariable chain rule, if I'm getting it right.