The Chain Rule for Partial Derivatives   

If w = f( x,y) and if x and y are functions of variables u and v, then the chain rule yields
 w
u
 =  w
x
  x
u
 +  w
y
  y
u
        and         w
v
 =  w
x
  x
v
 +  w
y
  y
v
That is, the chain rule for partial derivatives is a natural extension of the chain rule for ordinary derivatives.   

EXAMPLE 5    Find w/u and w/v when w = x2+xy and x = u2v, y = uv2.       

Solution: To begin with, the first partial derivatives of w = x2+xy are
 w
x
 = 2x+y,     w
y
 = x,
while the partial derivatives of x  and y with respect to u are
 x
u
 = 2uv,     y
u
 = v2
As a result, the chain rule says that
 w
u
 = ( 2x+y)  x
u
 +x   y
u
Substitution for x, y and their derivatives yields
 w
u
=
( 2u2v+uv2)2uv+u2v( v2)
=
4u3v2+3u2v3

        To evaluate w/v, we substitute the first partial derivatives to obtain
 w
v
 = ( 2x+y)  x
v
 +x   y
v
The partial derivatives of x and y with respect to v are
 x
v
 = u2,     y
v
 = 2uv
so that substitution for x, y and their derivatives yields
 w
v
=
( 2u2v+uv2)u2+u2v( 2uv)
=
2u4v+3u3v2

               

Check your Reading: How is example 5 related to w = u4v2+u3v3