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))⋅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))⋅y′(x).