5. THE CHAIN RULE
95
Implicit Di↵erentiation
Suppose F (x, y, z) = 0 implicitly defines a function z = f (x, y) where f is
@z
@z
di↵erentiable. Find
and
.
@x
@y
Let
@w
@x
@y
@z
= Fx
+ Fy
+ Fz .
@x
@x
@x
@x
@w
@x
@y
Now, since w = 0,
= 0. Also,
= 1 and
= 0. Then
@x
@x
@x
@z
0 = Fx + Fz .
@x
If Fz 6= 0,
@z
Fx
=
.
@x
Fz
Similarly,
@z
Fy
=
.
@y
Fz
w = F (x, y, z) =)
Theorem (Implicit Function Theorem). If Fx, Fy , and Fz are continuous
inside a sphere containing (a, b, c) where F (a, b, c) = 0 and Fz (a, b, c) 6= 0,
then F (x, y, z) = 0 implicitly defines z as a function of x and y near
(a, b, c).