The dependent variable is a function of the intermediate variables; these in turn are functions of the independent variables. Hence, the dependent variables is implicitly a function of the independent variables, but the dependence is hidden by the intermediates. The chain rule makes the dependence explicit!
If z=f(x,y) is a differentiable function of x and y, where x=g(t) and y=h(t) are differentiable functions of t, then
or
If z=f(x,y) is a differentiable function of x and y, where x=g(s,t) and y=h(s,t) are differentiable functions of s and t, then
and
The generalizations to multiple intermediates and multiple independent variables are relatively obvious.
Implicit differentiation is essentially an application of the chain rule. A function z is given implicitly as a function of x and y if there is there is an equation F(x,y,z)=0. If you like, the surface of the function z=f(x,y) is a level surface (equal to 0) of the function of three variables F!
If
- F be defined in a ball containing (a,b,c),
-
, and -
, 
, and 
are continuous inside the sphere.
and 
are
defined at (a,b) to be
and
The chain rule in multivariate functions is a straightforward generalization of the univariate chain rule: it just requires a lot more bookkeeping to keep track of things. Tree diagrams can help us keep our books.
Implicit derivatives are really just an application of the chain rule.