Chain Rule

Chain Rule

This rule is used to differentiate composite functions.

4For example y=(3x+2)

Chain Rule method

nIf y=u , where u is

expressed in terms x.

Then

dy dy du=

dx du dx

4y=(3x+2) , then u=3x+2 and 3du

dx

4

3 3

y=u

=4u 4(3 2)dy

xdu

3 3dy= =4(3x+2) 3=12(3x+2)

dx

dy du

du dx

4Differentiate y=(3x+2)

2Find the derivative of x +1

12 2y=(x +1)

2Let u=x +1

du=2x

dx1

2

1-

2

2

1 1

2 2 1

y u

dyu

du x

2 2

dy dy du 1 x= = .2x=

dx du dx 2 x +1 x +1

5

2

2Differentiate y= , 1.

1

xwith respect to x when x

x

2

2xLet u=

x +1 2 2 2 2

2 2 2 2 2 2

du (x +1)(2)-2x(2x) 2x +2-4x 2 2= =

dx (x +1) (x +1) ( 1)

x

x

5

44

2

y=u

dy 2x=5u =5

du x +1

4 2

2 2 2

dy dy du 2x 2 -2x= =5

dx du dx x +1 (x +1)

dy 2When x=1,then =5 (0)=0

dx 2

Differentiate the following exercises with respect to x:

3

4 5

4

2

1. (1+7x) Leaving Certificate Higher paper1 2005 no6(a)(i)

2. (x 1) Leaving Certificate Higher paper1 2002 no6(a)(i)

3. If y= 3 5, what is ?

2 54. If y= , hat is ?

15. If y= , hat is

2 5

dyx

dx

x dyw

x dx

dyw

x

?dx