13.5 Day 1 Chain Rule for
Functions of Several Variables
From Calc AB
Chain Rule:
dy dy du


dx du dx
If f g is the composite of y  f  u  and u  g  x  ,
then:

f
g

  fat u  g x  gat x
example:
f  x   sin x
g  x   x2  4
f   x   cos x
g  x   2x
Find:
f
g  at x  2
g  2  4  4  0
f   0  g   2
cos  0   2  2
1 4  4

We could also do it this way:
f  g  x    sin  x 2  4 
Evaluate at x=2
y  sin  x 2  4 
dy
 cos  x 2  4   2 x
dx
dy
 cos  22  4   2  2
dx
dy
 cos  0   4
dx
dy
4
dx

Example 1
How do you think we should do this problem?
Solve via substitution
Now we can do example 1 without first doing a substitution
Solution to Example 1
Example 2
Two objects are traveling in elliptical paths given
by the parametric equations given below. What
is the rate of change of the distance between the
objects when t = π?
Hint use the Chain rule:
Example 2 Solution
Solution: You can see that the distance s between the two objects is given by
Example 2 solution continued
Example 3
Find the answer by using substitution
Example 3 solution 1
We can now do example 3 without first having to do a substitution
There are some things that you
Just need to memorize!!!