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