2.4 The Chain Rule

Remember the composition of two functions?

))(( xgfgf =o

The chain rule is used when you have the composition of two functions.

[ ] )('))(('))(( xgxgfxgfdx

d=

Find y’ for y = (x2 + 1)3 f(x) = x3

g(x) = x2 + 1[ ] )('))(('))(( xgxgfxgf

dx

d=

y’ = 3(x2 + 1)2 (2x) = 6x(x2 + 1)2

Find y’ for y = (3x – 2x2)3

y’ = 3(3x – 2x2)2(3 – 4x)

Find f’(x) for 3 22 )2()( += xxf ( ) 322 2+= x

( ) )2(23

2)('

312 xxxf−

+=3 2 23

4

+=

x

x

Differentiate

( )( )232

7

−

−=

ttg rewritten as = -7(2t – 3)-2

g’(t) = 14(2t – 3)-3(2) 3)32(

28

−=

tDifferentiate

22 1)( xxxf −= rewritten as ( ) 2122 1 xx −

( ) ( ) ( ) ( )xxxxxxf 21212

1)('

2122122 −+−−⎟⎠

⎞⎜⎝

⎛=

−

( ) ( ) 2122123 121 xxxx −+−−=−

Factor

€

=x 1− x 2( )

−1 2−x 2 + 2 1− x 2

( )[ ]( )

2

2

1

32

x

xx

−

−=

Differentiate

3 2 4)(

+=

x

xxf rewritten as ( ) 312 4+

=x

x

Quotient Rule Bot * Top’ – Top * Bot’(Bot)2

322

322312

)4(

)2()4)(31()1()4()('

++−+

=−

xxxxx

xf ⎟⎠

⎞⎜⎝

⎛⋅3

3

322

3222312

)4(3

)4(2)4(3

++−+

=−

xxxx

( ) 342

2

43

12

+

+=

x

x

Differentiate2

2 3

13⎟⎠

⎞⎜⎝

⎛+−

=xx

y1

2 3

132' ⎟

⎠

⎞⎜⎝

⎛+−

=xx

y

€

x 2 + 3( ) 3( ) − 3x −1( ) 2x( )

x 2 + 3( )2

⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟

€

=2 3x −1( ) 3x 2 + 9 − 6x 2 + 2x( )

x 2 + 3( )3

€

=2 3x −1( ) −3x 2 + 2x + 9( )

x 2 + 3( )3

[ ]

[ ]dx

duuu

dx

ddx

duuu

dx

d

2

2

csccot

sectan

−=

= [ ]

[ ]dx

duuuu

dx

ddx

duuuu

dx

d

cotcsccsc

tansecsec

−=

=

Derivatives of Trigonometric Functions

[ ] [ ]dx

duuu

dx

d

dx

duuu

dx

dsincoscossin −==

Applying the Chain Rule to trigonometric functions

y = sin 2x

y = cos (x – 1)

y = tan 3x

y = cos (3x2)

y = cos2 3x

y’ = (cos 2x) (2) = 2 cos 2x

y’ = -sin (x – 1) (1) = -sin (x – 1)

y’ = sec2 3x (3) = 3 sec2 3x

y’ = -sin (3x2) (6x) = -6x sin (3x2)

rewritten as y = (cos 3x)2

y’ = 2(cos 3x)1 (-sin 3x) (3)

y’ = -6 cos 3x sin 3x

Differentiate

ttf 4sin)( = rewritten as 21)4(sin t

( ) 214sin2

1)(' −= ttf t4cos )4(

t

t

4sin

4cos2=