The Chain Rule
When gear A makes x turns, gear B makes u turns and gear C makes y turns., Chain rule
12
dydu
=3du
dx=
y turns ½ as fast as u
u turns 3 times as fast as x So y turns 3/2 as fast as x
dy dy dudx du dx
= g
Rates are multiplied
Consider a simple composite function: 6 10y x= −
( )2 3 5y x= −
If 3 5u x= −
then 2y u=
6 10y x= − 2y u= 3 5u x= −
6dydx
= 2dydu
= 3dudx
=
dy dy dudx du dx
= ⋅
6 2 3= ⋅
→
and another: 5 2y u= −
where 3u t=
( )then 5 3 2y t= −
3u t=
15dydt
= 5dydu
= 3dudt
=
dy dy dudt du dt
= ⋅
15 5 3= ⋅
( )5 3 2y t= −
15 2y t= −
5 2y u= −
→
and one more: 29 6 1y x x= + +
( )23 1y x= +
If 3 1u x= +
3 1u x= +
18 6dy xdx
= + 2dy udu
= 3dudx
=
dy dy dudx du dx
= ⋅
2y u=
2then y u=
29 6 1y x x= + +
( )2 3 1dy xdu
= +
6 2dy xdu
= +
( )18 6 6 2 3x x+ = + ⋅This pattern is called the chain rule.
→
The Chain Rule for composite functions
If y = f(u) and u = g(x) then y = f(g(x)) and
dydx
=dydui dudx multiply rates
dydx
=ddx
f (g(x)( ) = !f (g(x)) i !g (x)
multiply rates
dy dy dudx du dx
= ⋅Chain Rule:
If is the composite of and , then:
f go ( )y f u=
( )u g x=( ) ( ) at at xu g xf g f g=
ʹ′ ʹ′ ʹ′= ⋅o
example: ( ) sinf x x= ( ) 2 4g x x= − Find: ( ) at 2f g xʹ′ =o
( ) cosf x xʹ′ = ( ) 2g x xʹ′ = ( )2 4 4 0g = − =
( ) ( )0 2f gʹ′ ʹ′⋅
( ) ( )cos 0 2 2⋅ ⋅
1 4⋅ 4=→
Find the derivative (solutions to follow)
2 71) ( ) (3 5 )f x x x= −
2 232) ( ) ( 1)f x x= −
273) ( )
(2 3)f t
t−
=−
4) ( ) sin(2 )f x x=
25) ( ) tan( 1)f x x= −
Solutions 2 71) ( ) (3 5 )f x x x= −
2 232) ( ) ( 1)f x x= −
dy dy dudx du dx
= g
2
7 6
3 5 3 10
7
u x x x
y
du
dydu
x
u u
d→
→
= − = −
= =
6 3 10 )7 (dy ud
xx= −g
2 67(3 2 ) (3 10 )dy x x xdx
= − −
2
2 13 3
1 2
23
u x xd
dy u uydu
udx
−
= − =
= =
→
→
132
3(2 )d uy
dx
x−
= g
12 32 ( 1)
3(2 )dy
dxx x
−−= g
12 3
4
3( 1)
dy xdx
x
=
−
Solutions dy dy dudx du dx
= g
2 3
2 3 2
7 14
u t
y u
dd
d
u
du
x
uy− −
= − =
= − =
→
→
314 2udydx
−= g
33
2814(2 3) (2)(2 3)
dy tdx t
−= − =−
2 2
sin( ) cos( )
du
d
u x
yy
d
u udu
x→
→
= =
= =
2co 2 (2 )s( )dy ud
cos xx= =
22
73) ( ) 7(2 3)(2 3)
f t tt
−−= = − −
−
4) ( ) sin(2 )f x x=
25) ( ) tan( 1)f x x= −2
2
1 2
tan( ) sec ( )
u x x
y u dydu
dudx
u
= − =
= =
→
→
2 2 22 secsec ( )(2 )) ( 1dy u x xx
xd
= = −
Outside/Inside method of chain rule
( )( ( ) ( ( )) ( )dy d f g x f g x g xdx dx
ʹ′ ʹ′= = g
inside outside derivative of outside wrt inside
derivative of inside
think of g(x) = u
Outside/Inside method of chain rule example
( )1
2 33 1 ( ( )) ( )d x x f g x g xdx
⎛ ⎞⎜ ⎟ ʹ′ ʹ′− + =⎜ ⎟⎝ ⎠
g
inside
outside
derivative of outside wrt inside
derivative of inside
( ) ( ) ( )1 2
2 23 313 1 3 1 6 13
d x x x x xdx
−⎛ ⎞⎜ ⎟− + = − + −⎜ ⎟⎝ ⎠
g
( )2
2 3
6 1
3 3 1
x
x x
−
− +
Outside/Inside method of chain rule
( ) ( )33sin sin ( ( )) ( )d d f g x g xdx dx
θ θ ʹ′ ʹ′= = g
inside
outside
derivative of outside wrt inside
derivative of inside
( ) ( ) ( )3 2sin 3 sin sind ddx dx
θ θ θ=
23sin cosθ θ=
Outside/Inside method of chain rule
( )2csc( 3) ( ( )) ( )d f g x g xdx
θ ʹ′ ʹ′+ = g
inside outside derivative of outside wrt inside
derivative of inside
( )2 2 2csc( 3) csc( 3)cot( 3)2ddx
θ θ θ θ+ = − + +
2 22 csc( 3)cot( 3)θ θ θ− + +
( )1
2 2 2( ) 1f x x x= −
More derivatives with the chain rule 2 2( ) 1f x x x= −
( ) ( )1 1
2 2 2 22 2( ) 1 1d df x x x x xdx dx
ʹ′ = − + −
( ) ( )1 1
2 2 22 21( ) 1 ( 2 ) 1 22
f x x x x x x−
ʹ′ = − − + −
( )( )
132 2
12 2
( ) 1 2
1
xf x x x
x
−ʹ′ = + −
−
( )
( ) ( )
( ) ( ) ( )
1 12 22 23 3 2 3 3
1 1 1 12 2 2 22 2 2 2
1 2 1 (1 )2 2 2( )
1 1 1 1
x x xx x x x x x xf x
x x x x
− −− − + − − + −ʹ′ = + = =
− − − −
( )
3
12 2
3 2( )
1
x xf x
x
− +ʹ′ =
−
product
Simplify terms
Combine with common denominator
More derivatives with the chain rule 3
23 1( )
3xf xx−⎛ ⎞= ⎜ ⎟+⎝ ⎠
2
2 23 1 3 1( ) 3
3 3x d xf x
dxx x− −⎛ ⎞ ⎛ ⎞ʹ′ = ⎜ ⎟ ⎜ ⎟+ +⎝ ⎠ ⎝ ⎠
( )
2 2
2 22
3 1 ( 3)3 (3 1)(2 )33 3
x x x xx x
⎛ ⎞− + − −⎜ ⎟⎛ ⎞
⎜ ⎟ ⎜ ⎟+⎝ ⎠ ⎜ ⎟+⎝ ⎠
( )
2 2 2
2 22
3 1 (3 9) (6 2 )33 3
x x x xx x
⎛ ⎞− + − −⎜ ⎟⎛ ⎞
⎜ ⎟ ⎜ ⎟+⎝ ⎠ ⎜ ⎟+⎝ ⎠
( ) ( )
2 2 2
2 22
2 2
42
3(3 1) ( 3 2 9)3 1 3 9 6 233 3 3
x x x xx x
x x x
x
⎛ ⎞− + − +⎜ ⎟⎛ ⎞ =⎜ ⎟ ⎜ ⎟+⎝ ⎠ ⎜ ⎟+
⎝
− − + +
+⎠
Quotient rule