MAT 213Brief Calculus

Section 3.4

The Chain Rule

Consider…•An investor has been buying gold at a constant rate of 0.2 ounces per day. •If gold is currently worth $323.10 per ounce, how quickly is the total value of the investor’s gold increasing per day?

•How do we calculate this?•How could we use our derivative notation?

•Let g = gold, o = ounce, and t = time (measured in days

•What are the units?

62.642.010.323 dt

do

do

dg

dt

dg

Consider…

•A car is traveling at 100 feet per second.•What is the car’s speed in miles per hour?

•Hint: There are 5280 feet in a mile•How can we calculate this?•How can we write this in derivative notation?

•Let f = feet, m = mile, s = second, and h = hour

1

3600

5280

1

1

100

dh

ds

df

dm

ds

df

dh

dm

If y is a function of u and u is a function of x, then…

THE CHAIN RULE (Form 1)

dx

du

du

dy

dx

dy

Recall…

Composition of Functions

Results (outputs) of one process are used as inputs for another process.

NOTATION

Take the functions f(x) and g(x)

f(g(x)) = (f◦g)(x)

We can also have composition of more than two functions

f(g(h(k(x))))

Recall…

Composition of Functions

So we can think of composite functions as functions that have functions inside of them.

Composition of Functions

EXAMPLES

The function ln(x2) can be thought of as a composition of two other functions, ln x and x2, with the x2 being

INSIDE the ln function.

• Consider the function– We can “decompose” this function into two

functions we know how to take the derivative of– For example

– What are

– Now think of as

32 )3( xy

3)(and)( 23 xxguuufy

dx

du

du

dyand

dx

du

du

dy

dx

dy

dx

dy

The Derivative of a Composite Function

Let u=g(x) and y=f(u) be differentiable functions

So y=f(g(x)) is also differentiable

THE CHAIN RULE (Form 2)

The derivative of a composite function is the derivative of the outside function (leaving the inside function alone) times the derivative of the inside function

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

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

xgxgfxgf

xgxgfxgfdx

d

The Chain Rule…In Words

Derivative of the outside function with the inside

untouched

Derivative of the inside

function

Derivative of a composite

function= x

• The Chain Rule allows us to generalize some of our previous rules

11

( ) ( )

( ) ( ) '( )

1 '( )ln ln ( )

( )

'( )

n nn n

x x f x f x

d dx nx f x n f x f x

dx dxd d f x

x f xdx x dx f x

d de e e e f x

dx dx

“Simple” Rules Generalized (Chain) Rules

The Chain Rule

Examples

23 1

2 2

2

( )

3( )

(4 1)

( ) 5 8

mh m e

v tt

g x x - x

)87ln()( 2 xxxf3

3

0.5

( ) 5

( ) ln(2 )

23( )

1 2

t

x

t

f t

k x

P te