Rule: chain rule for a composition of three functions
For all values of
x for which the function is differentiable, if
then
In other words, we are applying the chain rule twice.
Notice that the derivative of the composition of three functions has three parts. (Similarly, the derivative of the composition of four functions has four parts, and so on.) Also,
remember, we can always work from the outside in, taking one derivative at a time.
At this point, we present a very informal proof of the chain rule. For simplicity’s sake we ignore certain issues: For example, we assume that
for
in some open interval containing
We begin by applying the limit definition of the derivative to the function
to obtain
Rewriting, we obtain
Although it is clear that
it is not obvious that
To see that this is true, first recall that since
g is differentiable at
is also continuous at
Thus,
Next, make the substitution
and
and use change of variables in the limit to obtain
As with other derivatives that we have seen, we can express the chain rule using Leibniz’s notation. This notation for the chain rule is used heavily in physics applications.
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