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We’ve been using the standard chain rule for functions of
one variable throughout the last couple of sections. It’s now time to extend the chain rule out to
more complicated situations. Before we
actually do that let’s first review the notation for the chain rule for
functions of one variable.
The notation that’s probably familiar to most people is the
following.
There is an alternate notation however that while probably
not used much in Calculus I is more convenient at this point because it will
match up with the notation that we are going to be using in this section. Here it is.
Notice that the derivative 
really does make sense here since if we were
to plug in for x then y really would be a function of t.
One way to remember this form of the chain rule is to note that if we
think of the two derivatives on the right side as fractions the dx’s will cancel to get the same
derivative on both sides.
Okay, now that we’ve got that out of the way let’s move into
the more complicated chain rules that we are liable to run across in this
course.
As with many topics in multivariable calculus, there are in
fact many different formulas depending upon the number of variables that we’re
dealing with. So, let’s start this
discussion off with a function of two variables, . From this point there are still many
different possibilities that we can look at.
We will be looking at two distinct cases prior to generalizing the whole
idea out.
Case 1 : ,
,
and compute .
This case is analogous to the standard chain rule from
Calculus I that we looked at above. In
this case we are going to compute an ordinary derivative since z really would be a function of t only if we were to substitute in for x and y.
The chain rule for this case is,
So, basically what we’re doing here is differentiating f with respect to each variable in it
and then multiplying each of these by the derivative of that variable with
respect to t. The final step is to then add all this up.
Let’s take a look at a couple of examples.
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Example 1 Compute
for each of the following.
(a) ,
,
[Solution]
(b) ,
,
[Solution]
Solution
(a) ,
,
There really isn’t all that much to do here other than
using the formula.
So, technically we’ve computed the derivative. However, we should probably go ahead and
substitute in for x and y as well at this point since we’ve
already got t’s in the
derivative. Doing this gives,
Note that in this case it might actually have been easier
to just substitute in for x and y in the original function and just
compute the derivative as we normally would.
For comparisons sake let’s do that.
The same result for less work. Note however, that often it will actually
be more work to do the substitution first.
[Return to Problems]
(b) ,
,
Okay, in this case it would almost definitely be more work
to do the substitution first so we’ll use the chain rule first and then
substitute.
Note that sometimes, because of the significant mess of
the final answer, we will only simplify the first step a little and leave the
answer in terms of x, y, and t. This is dependent upon
the situation, class and instructor however and for this class we will pretty
much always be substituting in for x
and y.
[Return to Problems]
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Now, there is a special case that we should take a quick
look at before moving on to the next case.
Let’s suppose that we have the following situation,
In this case the chain rule for becomes,
In the first term we are using the fact that,
Let’s take a quick look at an example.
Now let’s take a look at the second case.
Case 2 : ,
,
and compute and .
In this case if we were to substitute in for x and y we would get that z is
a function of s and t and so it makes sense that we would be
computing partial derivatives here and that there would be two of them.
Here is the chain rule for both of these cases.
So, not surprisingly, these are very similar to the first
case that we looked at. Here is a quick
example of this kind of chain rule.
Okay, now that we’ve seen a couple of cases for the chain
rule let’s see the general version of the chain rule.
Chain Rule
Suppose that z is
a function of n variables, ,
and that each of these variables are in turn functions of m variables, . Then for any variable ,
we have the following,