1. 7.7 The Chain Rule
A. What’s a Composition of Functions
B. The Chain Rule
C. When a Problem Uses More than
One Rule
2. A. What’s a composition of
functions?
• You’ll have an “inside function” and an
“outside function” sort of.
• We will call the outside function f and the
inside function g.
• When you plug the function g INTO f, you
get the original composition of functions.
• You will understand these words better if I
show you….
3. f ( x ) = x and g ( x ) = 4 − x.
2
Find f ( g ( x ) ).
This means PLUG g INTO f.
2
Skeleton of f : ( )
Insert g : ( 4 − x )
2
This is a composition of two functions!
4.
5. Consider ( 4 − x ) .
2
You canNOT just use the power rule to bring down the 2
and get 2( 4 − x ) . That would be WRONG.
1
Why can' t we do this? If it were x 2 , we could do it!
The base isn' t just plain x this time. Instead
the base is ( 4 - x ).
This will be a job for.....THE CHAIN RULE!
The first step is identifying an inside function g and an
outside function f such that f(g(x)) = ( 4 - x ) .
2
6. B. The Chain Rule
d
( f ( g ) ) = f ′( g ) ⋅ g ′ means that we CAN use the power rule
dx
in that way AS LONG AS we multiply it by the derivative of
the inside function, which I might refer to as its " tail."
d
dx
( )
( 4 − x ) 2 = 2( 4 − x )1 ⋅ ( − 1)
In this one, the derivative of the inside function (4 - x) is (-1).
