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D2. Differentiation Rules Continued
If 
, then 
can be said to be a function of 
. If you let 
then:
where
is now a function of 
, and is a function of . The new variable, , is the link between the two expressions.
To differentiate the expression with respect to , you would first need to expand the bracket. In this particular case, it would not be too difficult, however if the expression was 
, it would involve much more work. The chain rule allows us to differentiate such expressions more easily.
The Chain Rule
If 
where 
, and 
and 
are differentiable,
then is a differentiable function of and:
Using the example above of ,
where
Differentiating with respect to gives: Differentiating with respect to gives:
By the chain rule,
=
Ex.1. Find 
if:
a. 
b. 
c. 
d. 
Ex.2. Differentiate the following functions:
a. 
b. 
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