12.5:  Chain Rules for Functions of Several Variables

1.	Theorem 12.6:  Chain Rule:  One Independent Variable

Let w = f(x, y), where f is a differentiable function of x and y.  If x = g(t) and y = h(t), where g and h are differentiable functions of t, then w is a differentiable function of t, and

	The chain rule can be extended to any number of variables.  

2.	Theorem 12.7:  Chain Rule:  Two Independent Variables

Let w = f(x, y), where f is a differentiable function of x and y.  If x = g(s,t) and y = h(s,t) such that the first partials all exist, then 

  and  

Again, this chain rule can be extended to any number of variables.  

3.	Theorem 12.8:  Chain Rule:  Implicit Partial Differentiation

If the equation F(x, y) = 0 defines y implicitly as a differentiable function of x then,

If the equation F(x,y,z) = 0 defines z implicitly as a differentiable function of x and y then

  and  

	This theorem can be extended to differentiable functions defined implicitly with any number of variables.