Chain rule: special case 1

Introduction: By the use of the chain rule we can determine the derivative of a composite function. Such a composite function depends on two variables, where the variables themselves depend on a third variable:
$$Z(t) = z(x(t),y(t)).$$
The function $Z(t)$ is a function of one variable, but is composed of the functions $z(x,y)$, $x(t)$ and $y(t)$ where $z(x,y)$ is a function of two variables. For such a composite function we can use the chain rule, which gives the following.

Theorem: If $Z(t) = z(x(t),y(t))$, then
$$Z'(t) = z'_x(x(t),y(t)) \cdot x'(t) + z'_y(x(t),y(t)) \cdot y'(t).$$