$f(x,y,z,t)$, $x(t)$, $y(x,t,s)$, $z(y,x)$
So differentiating $\frac{\partial f}{\partial t}= \frac{\partial f}{\partial x}\frac{\partial x}{\partial t}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial t}+\frac{\partial f}{\partial z}\frac{\partial z}{\partial t}+\frac{\partial f}{\partial t}$
then differentiating y and z with respect to $t$, $\frac{\partial f}{\partial t}= \frac{\partial f}{\partial x}\frac{\partial x}{\partial t}+\frac{\partial f}{\partial y}[\frac{\partial y}{\partial x}\frac{\partial x}{\partial t}+\frac{\partial y}{\partial t}]+\frac{\partial f}{\partial z}[\frac{\partial z}{\partial x}\frac{\partial x}{\partial t}{\frac{\partial z}{\partial y}[\frac{\partial y}{\partial x}\frac{\partial x}{\partial t}+\frac{\partial y}{\partial t}}]]+\frac{\partial f}{\partial t}$
Is this correct?