The Chain Rule , finding the derivative of a composition.
A composition of functions is a function , u(x) inside another function, f(u), making a
function of x, f(u(x)).
For example
f (x) 
The inside function is
x  3 x  20
3
u ( x )  x  3 x  20 .
3
The outside function is f ( u )  u
If x changes , then u changes and then f changes.
Consider two linear functions,
u( x) 
1
f ( u )  3 u The composition is
x
2
1
f ( u ( x ))  3 
x 
2
3
x
2
The slopes multiplied. The chain rule says the same is true for nonlinear functions.
The Chain Rule:
df
df du

dx
Examples in which
 f ' (u )u ' ( x )
du dx
u( x)  x  3 :
2
a ) h ( x )  ( x  3)
2
5
h ' ( x )  5( x  3) ( 2 x )
2
4
Here h ( x )  f ( u ( x )) where f ( u )  u
5
f ' ( u ) u ' ( x )  5u ( 2 x )  5( x  3) ( 2 x )
4
b) f ( x) 
1
x 3
2
2
 ( x  3)
2
2
1
 2x
2
f ' ( x )   ( x  3) 2 x 
4
x
2
 3
2
f(x) in part b could also be done using the quotient rule.
More examples : Find the derivative of each function.
1)
f ( x )  ( x  1)
2)
g ( x )  ( 2 x  5)
3)
h ( x )  ( x  1)
4)
f (x) 
3
3
2
x
( x  1)
2
2
4
5)
g(x) 
2x
3
5x  7
2
6) A snowball is a perfect sphere with volume given by V
(r) 
4
 r
3
cubic cm.
3
Find the rate of change of the volume with respect to the radius.
If the radius is decreasing by 2 cm per min. due to melting, how fast is the volume
decreasing when the radius is 30 cm?
7) The revenue as a function of x=number of units sold in the month is given by
R ( x )   0 . 2 x  40 x . If the quantity sold is decreasing by 12 units per month,
2
at what rate is revenue changing with respect to t=time in months if x=150?