In this Khan Academy video (https://youtu.be/YT6XwkcPcsw?t=138), Sal takes partial derivatives of several dependent variables.
When taking the full derivative of $Q(x,\ y,\ z(x,\ y))$ with respect to $x$, one must sum all paths through variable dependencies which lead to $x$, yielding $\frac{dQ}{dx} = \frac{∂Q}{∂x} + \frac{∂Q}{∂z}\frac{dz}{dx}$. However, the video treats partial derivatives in the same way that one normally treats full derivatives, contrary to my prior understanding. In fact, the partial derivative $\frac{∂Q}{∂x}$ appears in the formula for the full derivative $\frac{dQ}{dx}$, so when the same formula is applied to partial derivatives, it looks like he is saying, $\frac{∂Q}{∂x} = \frac{∂Q}{∂x} + \frac{∂Q}{∂z}\frac{dz}{dx}$, which is nonsense if $Q$ depends on $z$ nontrivially. What is going on here?