Atmospheric temperature depends on position and time. If we denote position by the three spatial coordinates $x, y,$ and $z$ (measured in kilometers) and time by $t$ (measured in hours), then the temperature at a given location and time is given by $T = f(x,y,z,t)$. Suppose that a weather balloon measures the temperature at a certain location and time. The balloon moves on a path with parametric equations $$x=t, \quad y=3t, \quad z=3t-t^2$$ Find the recorded temperature and its rate of change at time $t=2$ if $$T(x,y,z,t) = \frac{xy(2+t)}{1+z}$$

Consider $z=\sin(3x-y)$ and $x=s^2+t^2$ and $y=st$

Suppose that $z$ depends on $u, v$ and $r$; $u$ and $v$ depend on $x, y$ and $r$; $r$ depends on $x$ and $y$.