2.4 The Chain Rule

"I recoil with dismay and horror at this lamentable plague of functions which do not have derivatives. -- Charles Hermite

Objectives:

To find derivatives using the chain rule

Two ways

Find the derivative of 2)56( xy

Power Product

A third way…

Chain rule

y’ = deriv of outside (keep inside same) times deriv of inside

Example:

)('))((''))((

xgxgfyxgfy

2)56( xy

A fourth way

Another form of the chain rule:

Example

')('

)()())((

uufdxdy

xguufyxgfy

2)56( xy

Example

2)12( xy

Another example

32 )1( xy

What about trig functions?

Think of u substitution

xxf 2sin)(

You try:32 )23( xxy 5)(sin xy

)sin( 5xy 4)(Bobdxd

More challenging examples…

3 22 )1( xy

Another one

2)32(7)(

xxg What rules will we use?

Yet one more…22 1 xxy

Review:

Chain Rule:

)()('

)(1 quantityofderivquantitykny

quantitykyn

n

2 more examples

3 2 4

x

xy

Last one2

2 313

xxy

Trig Chain Rule

Deriv of trig (keep angle same) deriv of angle

Be careful of double chain rule…

Example

xy 2sin41 2

Another example

ty 4sin3

Three types…

)3cos( 2xy 2)3cos( xy

2))3(cos( xy