A conceptual question on implicit partial derivatives in maltivariate situations

Question

Say q is a function of two variables m and H: $$q = f_{1}(m,H)$$ Further, m and H are related to a third variable L: $$L = f_{2}(m,H)$$ I believe that, using the idea of the total derivative, it is correct to say: $$dq = \frac{\partial q}{\partial m}dm + \frac{\partial q}{\partial H}dH$$ Similarly, $dm = \frac{\partial m}{\partial L}dL$ and $dH = \frac{\partial H}{\partial L}dL$ should also be correct. Given this background, are the following equations correct? $$\frac{dq}{dm} = \frac{\partial q}{\partial m} + \frac{\partial q}{\partial H}\frac{dH}{dm}$$ or $$ \frac{dq}{dm} = \frac{\partial q}{\partial m} + \frac{\partial q}{\partial H} \frac{\frac{\partial H}{\partial L}}{\frac{\partial m}{\partial L}}$$ If these are correct, then are they circular? In other words, would $\frac{dq}{dm}$ be obtainable directly from the first equation too, with the same result? Any help and guidance would be greatly appreciated.