I'm working through a particular problem that is giving me inconsistent results given the way I approach the problem, and I would like to confirm that I am appropriately understanding the chain rule in the case where my function, call it $z$, is a function of some $x(t)$ and some $y(x(t))$. That is, the function $z$ is a function of two variables, the second of which is a function of the first, or
$$z = f(x(t),y(x(t))).$$
I'm differentiating with respect to $t$ by applying the following rule:
$$\frac{\partial z}{\partial t} = \frac{\partial z}{\partial x}\frac{\partial x}{\partial t} + \frac{\partial z}{\partial y}\frac{\partial y}{\partial x}\frac{\partial x}{\partial t} = \frac{\partial x}{\partial t}\biggl[\frac{\partial z}{\partial x} + \frac{\partial z}{\partial y}\frac{\partial y}{\partial x} \biggr].$$
Is this the correct application of the chain rule in this case?