ACOW RATES OF CHANGE AND DERIVATIVES MODULE
Updated 5/28/2016
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Activity C3 – The Chain Rule
Functions can be combined in various ways (sum, difference, product, quotient) to create
new functions. The composition of functions is another way to form a new function.
You can think of the composition of functions as substituting one function into another.
To help understand this concept, it is helpful to view some examples (RS2).
Given f ( x)  x51 and g ( x)  x 2  2 x  5 , find
1. ( f g )( x)  f ( g ( x))
2. ( g f )( x)  g ( f ( x))
It is often helpful to view the composition of two functions as an inner function and an
outer function. When viewed in this manner, we can easily decompose a function into
two functions. To help understand how to decompose a function, it will be helpful to
view some examples(RS3).
3. Given h( x)  ( x  2)2  1 and h( x)  ( f g )( x) , which of the functions below would
correctly represent f(x) and g(x)?
a) f ( x)  x  2, g ( x)  x 2
b) f ( x)  x 2  1, g ( x)  x  2
c) f ( x)  x 2 , g ( x)  x 2  1
d) f ( x)  x  2, g ( x)  x 2  1
Notice that both of the functions below are the composition of two functions. For
example, we can decompose g(x) in the two functions h( x)  x 2 2 x and p ( x)  x ,
then define g ( x)  ( p h)( x) .
g ( x)  x 2  2 x
f ( x)   4 x1/ 3  7 
99
Because g(x) is the composition of two functions, we must apply The Chain Rule(RS4) to
find the derivative of g(x). Applying the chain rule to g(x) we can find g’(x)(RS5).
Use the chain rule to find the derivatives of the following functions.
4. h( x)  (2 x3  4 x)80
5. q( x)  3 x1.2  5 x 4
ACOW RATES OF CHANGE AND DERIVATIVES MODULE
Updated 5/28/2016
Page 2 of 2
Another rule that is commonly used in place of the chain rule is the general power
rule(RS6). To demonstrate this let g ( x)  x 2  2 x and apply the general power rule to
find g '( x ) (RS7).
Find the derivatives of the following functions.
6. f ( x)   7 x 2  8 x  1
41
7. g ( x)  3 x3  8 x  5
8. h( x) 
 2.1x
2
2
 5.7 x  8 
2