The Chain Rule

• Rule for finding the derivative of a composition of two functions.

• If y is a function of u and u is a function of x, then y is a function of x.

• The chain rule tell us how to find the derivative of y with respect to x

Example

• A factory produces 50 items per hour and the manufacturing costs are $0.25 per item. What is the manufacturing cost per hour at the factory?

• Manufacturing costs are $12.50 per hour.

• Answer comes from multiplying the rates:

(0.25 $/item)(50 items/hour) = 12.50 $/hour

• That’s the chain rule!

• Let be the manufacturing cost in dollars.

• Let be the number of items produced.

• Let be the time in hours

• cost per hour = (cost / item)(items / hour)

• In terms of derivatives:

y

dydx

=dydu

£dudx

u

x

The Chain Rule

• Suppose is a differentiable

function of and is a

differentiable function of

• Then is a differentiable

function of

)(ufy

))(( xgfy

)(xgu ux

x

The Chain Rule

• We have

• The derivative of is given by

equivalently,

))(( xgfy

dydx

=dydu

£dudx

dydx

= f 0(g(x))g0(x)

))(()( xgfufy )(xgu

then

• To use: Think of as an “outside function”

applied to an “inside function”

• Use a new variable for the “inside function”

• Rewrite the “outside function” in terms of the new variable

• Take the derivatives and multiply

• Rewrite all in terms of the original variable

• Tricky part is deciding what is the “inside function” and “outside function”

y= f (g(x)) dydx

= f 0(g(x))g0(x)

g(x)fy

• Find the derivative of

• Let “inside function”

• Then “outside function”

where

y= (2x2+1)2

u= 2x2+1y= u2

y= f (u)

(2u)(4x) = 2(2x2+1)(4x) =

16x3+8x

dydx = f 0(u)(u0) =

f (x) =u2;u=2x2+1

f0(x) =2u ; u0 =4x

• Without the chain rule:

• This agrees with our previous calculation!

y= (2x2+1)2 = 4x4+4x2+1

ddx

¡4x4+4x2+1

¢=

16x3+8x

• Find the derivative of

• Let “inside function”

• Then “outside function”

y =p3x2+x

u= 3x2+x

dydx = 1

2(u)¡ 1=2(u0) =

6x+12p3x2+x

2/1uuy

12(3x

2+x)¡ 1=2(6x+1) =

The General Power Rule

• Combine the Power Rule and Chain Rule

• If is differentiable and is any real number, then

)()()( 1 xgdx

dxgnxg

dx

d nn

)(xg n

')()'( 1 uunu nn

• Apply to

• Then

where

• By the Generalized Power Rule

nuxxy 2/12 )3(

y =p3x2+x

dydx = 1

2(3x2+x)¡ 1=2(6x+1) =

')()'( 1 uunu nn

2/1,3 2 nxxu

Sometimes the chain rule must be combined with the product or quotient rule. For example, to differentiate

we use the product rule, but we must use the chain rule to take the derivative of

(x3+1)4(2x+2)

(x3+1)4