Introduction: A composite function can also depend on two variables, where one of the variables depends on the other:
$$Z(x) = z(x,y(x)).$$
The function $Z(x)$ is a function of one variable, but is composed of the functions $z(x,y)$ and $y(x)$ 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 $Z(x) = z(x,y(x))$, then
$$Z'(x) = z'_x(x,y(x)) + z'_y(x,y(x)) \cdot y'(x).$$
$$Z(x) = z(x,y(x)).$$
The function $Z(x)$ is a function of one variable, but is composed of the functions $z(x,y)$ and $y(x)$ 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 $Z(x) = z(x,y(x))$, then
$$Z'(x) = z'_x(x,y(x)) + z'_y(x,y(x)) \cdot y'(x).$$