Calc 3 Lecture Notes
Section 12.5
Page 1 of 5
Section 12.5: The Chain Rule
Big idea: To take the derivative of a function of more than one variable when each of the
variables is a function of other variables requires a special application of the chain rule.
Big skill: You should be able to apply the chain rule to functions of several variables.
Theorem 5.1: The Chain Rule
If z  f  x t  , y t   , where x(t) and y(t) are differentiable and f(x, y) is a differentiable function
of x and y, then
dz d
f
dx f
dy f dx f dy
  f  x  t  , y  t     x  t  , y  t     x  t  , y  t   

dt dt
x
dt y
dt x dt y dt
Proof (using the definition of differentiability):
Tree Diagram to sort out the dependencies:
Calc 3 Lecture Notes
Section 12.5
Page 2 of 5
Practice:
1. Let z  f  x, y   x2e y , x  t   t 2  1, and y  t   sin  t  . Find
dz
. Notice that we could
dt
have avoided Theorem 5.1 by substituting first.
2. The Cobb-Douglas production function models production P as a function of available
capital k (in M$), and the available labor L (in thousands of peons) as
P  k , L   20k 0.25 L0.75 . Find the instantaneous rate of change of production when k is
changing at a rate of +$500,000/yr and L is decreasing by 100 peons per year.
Calc 3 Lecture Notes
Section 12.5
Page 3 of 5
Q: Now what happens when each of the variables of a two-variable function are themselves
functions of two variables?
A: The chain rule still applies; you just have to be a bit more careful.
Theorem 5.2: The Chain Rule
If z  f  x, y  , where f(x, y) is a differentiable function of x and y, and where x  x  s, t  and
y  y  s, t  both have first-order partial derivatives, then the following chain rules apply:
z f x f y
z f x f y
and




s x s y s
t x t y t
Tree Diagram:
Practice:
3. Let z  f  x, y  
x2
z
 y 2 , x  t , u   2u cos t  , and y  t , u   u sin  t  . Find
and
t
4
z
. Notice that we could have avoided Theorem 5.2 by substituting first.
u
Calc 3 Lecture Notes
Section 12.5
Page 4 of 5
4. Let a differentiable function f  x, y  be parameterized in terms of polar coordinates:
x  r,   r cos   , y  r,   r sin   . Find fr and frr.
Calc 3 Lecture Notes
Section 12.5
Page 5 of 5
Dimensionless Variables…
Are easier to work with once you get the hang of them.
Practice:
v0 2 R
5. Rewrite the rocket height equation h 
using dimensionless variables. The
19.6 R  v0 2
equation uses metric units, R is the earth’s radius, v0 is the launch velocity of the rocket,
and h is the height it attains. Notice how much easier it is to graph.
Implicit Differentiation
If w  F  x, y, z   0 , then
F
F z
z
 x ,
  y , etc.
x
Fz y
Fz
Practice:
6. F  x, y,   xy 2  e xy cos  x   0
7. F  x, y, z   xy 2  z 3  sin  xyz   0