# Partial derivative of $v$ w.r.t. $x$ in Lagrangian dynamics [duplicate]

In Lagrangian dynamics, when using the Lagrangian thus:

$$\frac{d}{dt}(\frac{\partial \mathcal{L} }{\partial \dot{q_j}})- \frac{\partial \mathcal{L} }{\partial q_j} = 0$$

often we get terms such as the one below which result in zero:

example 1: $$\frac{\partial \dot{q}}{\partial q} = 0$$ example 2: $$\frac{\partial q}{\partial \dot{q}} = 0$$

I wasn't sure why but this is the closest I have come to reasoning it out:

$$\frac{\partial \dot{q}}{\partial q} = \frac{\partial \dot{q}}{\partial t}\frac{\partial t}{\partial q} = 0$$

because

$$\frac{\partial t}{\partial q} = 0$$

and similar for example 2. Is this correct?