2.4 Chain Rule

Chain Rule• If y=f(u) is a differentiable function of u and u=g(x) is a

differentiable function of x then y=f(g(x)) is a differentiable function of x and

or

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

dy dy du

dx du dxdf g x f g x g x

dx

Try chain rule• Find the derivative of

2( )

3 1f x

x

Answer• Use chain rule

1

2

2

2

( ) 2(3 1)

'( ) 2( 1)(3 1) (3)

6(3 1)

6

(3 1)

f x x

f x x

x

x

Try chain rule• Find the derivative of

3( ) ( 2)f x x

Answer• Use chain rule

2

2

'( ) 3( 2) (1)

3( 2)

f x x

x

Chain rule with trig• Find the derivative of

( ) sin(2 )f x x

Answer• Use chain rule

'( ) cos(2 )(2)

2cos(2 )

f x x

x

Chain rule and quotient rule• Find the derivative of

323 2( )

2 3

xg x

x

Answer• Use chain and quotient rules

22 2

2

2 2 2 2

4

2 2 2

4

3 2 (2 3)(6 ) (3 2)(2)'( ) 3

2 3 (2 3)

3(3 2) (12 18 6 4)

(2 3)

3(3 2) (6 18 4)

(2 3)

x x x xg x

x x

x x x x

x

x x x

x

Trig and chain rule

2 2

[sin ] (cos ) ' [cos ] (sin ) '

[tan ] (sec ) ' [cot ] (csc ) '

[sec ] (sec tan ) ' [csc ] (csc cot ) '

d du u u u u u

dx dxd d

u u u u u udx dxd d

u u u u u u u udx dx

Chain rule again• Find the derivative of y = sin πx

Answer• Use chain rule

' cos ( )

cos( )

y x

x

Derivative again• Find the derivative of

2( ) 5cosg t t

Answer• Use chain rule twice!!

'( ) 5(2)cos ( sin )( )

10 sin( )cos( )

g t t t

t t

Tangent line!• Find the equation of the tangent line to

3( ) 2 tan

@( ,2)4

f x x

Answer• Use chain rule and point-slope

2 2

2 2

2 24 4 4

2 2

4

'( ) 2(3) tan (sec )

6 tan sec

'( ) 6(tan( )) (sec( ))

6(1) ( 2) 12

2 12( ( ))

f x x x

x x

f

y x