Section 15.5
The Chain Rule
“How to differentiate compositions of functions”
Recall that in single variable calculus, if
f
(
g
(
x
)) is a composition of
functions, then
df
(
g
(
x
))
dx
=
f
′
(
g
(
x
))
g
′
(
x
). For functions of more than one
variable, this is no longer directly the case since there are many different
variables we could differentiate with respect to. We need to generalize
the idea of the chain rule to functions of more than one variable.
1.
Tree Diagram
In order to develop the chain rule, we need a new idea called a tree
diagram. We set it up as follows:
•
Suppose
f
is a function of
n
variables, call them
x
1
, . . . , x
n
.
Suppose in addition, the variables
x
1
, . . . , x
n
are also func
tions of
m
other variables, call them
u
1
, . . . , u
m
(and we could
continue letting
u
1
, . . . , u
m
depend upon other variables etc).
Then we can construct a diagram which reflects these depen
dencies.
•
On the top row of the diagram, we write
f
•
On the the next row we write the variables
f
directly depends
upon (
x
1
, . . . , x
n
).
•
On the the next row the variables
x
1
, . . . , x
n
depend upon
(
u
1
, . . . , u
m
).
•
We continue to do this until all variables are written down.
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 Fall '08
 Stefanov
 Calculus, Chain Rule, Derivative, The Chain Rule, Tree Diagram, 1.8cm

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