homework and exercises - Proving one of the canonical Hamilton equations

I have a homework question asking me to prove that $\frac{\partial \mathcal{H}}{\partial t}=-\frac{\partial\mathcal{L}}{\partial t}$, and the hint is to first consider a system with one degree of freedom where the equation for the Hamiltonian simplifies to $\mathcal{H}(q,p,t)=p\dot{q}(q,p,t)-\mathcal{L}(q,\dot{q}(q,p,t),t)$. Now I tried to take the partial derivative with respect to time but I'm not sure that I'm doing it right, for my first step I get $$\frac{\partial\mathcal{H}}{\partial t}=p\frac{\partial \dot{q}(q,p,t)}{\partial t}\frac{\partial q}{\partial t}-\frac{\partial \mathcal{L}(q,\dot{q}(q,p,t),t)}{\partial t}\frac{\partial q}{\partial t}\frac{\partial \dot{q}(q,p,t)}{\partial t}\frac{\partial q}{\partial t}$$
I'll spare you the details of simplifying, but I get stuck at $$\frac{\partial\mathcal{H}}{\partial t}=-\frac{\partial \mathcal{L}}{\partial t}\Big(\frac{\partial ^2q}{\partial t^2}\frac{\partial\dot{q}}{\partial t}-\frac{\partial q}{\partial t}\Big)$$ So in theory, the stuff in the parenthesis should equal one if I'm doing this right, but I'm unsure if I am. Any help would be greatly appreciated!