The Chain Rule for Functions of Several Variables

 1 In a Nut Shell:  Recall that the chain rule for a function, y, of one independent variable, x, was very important in that it enabled you to replace the derivative of a complicated function with a product of simpler derivatives, called the “chain” rule.   Recall the Chain Rule:   If   u  =  u(x)  and   x = x(t),   then  du/dt  =  [du/dx][dx/dt]   Now consider a function,  u(x,y)   (which has continuous, first-order derivatives) and that       x  =  g(t)  and   y  =  h(t)  are differentiable functions.  Then  u  is a differentiable function of  t  .   Terminology:   u   is the dependent variable,      x and  y   are intermediate variables   and   t   is the independent variable   Suppose  u  =  u(x,y) has continuous first-order derivatives.   Then          du/dt   =   [∂u/ ∂x] dx/dt    +  [∂u/ ∂y] dy/dt 2 It can be helpful to construct a “tree diagram” to illustrate the chain rule structure.                           Tree diagram for  w = w(x,y)  where  x = x(t)  and  y = y(t).     It follows then that the chain rule for   dw/dt  is given as above by the expression:               dw/dt   =   [∂w/ ∂x] dx/dt    +  [∂w/ ∂y] dy/dt   Click here to continue discussion of the multivariable chain rule.