Let $a = bc$. Then $b = a/c$. From the first equation, we also have $\frac{\partial a}{\partial c} = b$. Equating, $\frac{\partial a}{\partial c} = b = a/c$, or $\frac{\partial a}{\partial c} = a/c$, and any partial derivative can be expressed a fraction of its constituents.
Is there something wrong with this proof? If so, what? This looks like such a simplification that I'm convinced it must be wrong, but I can't find a counterexample.
You've assumed that $a$ is a scalar multiple of $c$ and therefore (assuming both are differentiable) $a'$ is going to be the same scalar multiple of $c'$ by elementary properties of derivatives. It's not a powerful result, because you put some pretty extreme restrictions on $a$ and $c$, namely that they are scalar multiples.