3.3 Differentiation Rules

Colorado National MonumentPhoto by Vickie Kelly, 2003

Created by Greg Kelly, Hanford High School, Richland, WashingtonRevised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts

If the derivative of a function is its slope, then for a constant function, the derivative must be zero for all x.

0dc

dx

example: 3y

(3) or (3) 0d

ydx

The derivative of a constant function is zero.

We saw that if , .2y x 2dy

xdx

This is part of a larger pattern!

1n ndx nx

dx

examples:

f x x

01 1f x x

8y x

78y x

“power rule”

( )

!

d

dxd

dxd

dxwhere is some

u

u

u

u

functi n

c

f x

c

o

c

o

examples:

( )ncd

xdx

constant multiple rule:

5 5(7 ) 7 ( )d d

x xdx dx

47 5x 435x

( )ncdx

dx1ncnx

(Each term is treated separately)

sum and difference rules:

ud d d

dx d x

vv

x du ud d d

dx d x

vv

x du

4 12y x x

y

4 23 2 9y x x dy

dx

12 4x

!fu

whe

nct

re and

ions

reu v

of x

a

034x 312x

Example:Find the horizontal tangents of: 4 22 2y x x

34 4 0dy

x xdx

Horizontal tangents: (slope of y) = (derivative of y) = zero.

34 4 0x x

3 0x x

2 1 0x x

1 1 0x x x

0, 1, 1x

Substitute these x values into the original equation to generate coordinate pairs: (0, 2), (-1, 1) and (1,1)

2 0( 0)

1 0( 1)

1 0( 1)

y x

y x

y x

(So we expect to see two horizontal tangents, intersecting the curve at three points.)

… and write tangent lines:

4 22 2y x x

4 22 2y x x

2y

4 22 2y x x

2y

1y

4 22 2y x x

First derivative y’ (slope) is zero at: 1, 0, 1x

34 4dy

x xdx

…where y has slopes of zeroThe derivative of y is zero

at x= -1, 0, 1…

product rule:

d du dvu v v u

dx dx dx

Notice: the derivative of a product is not just the product of the two derivatives. (Our authors write this rule in the other order!)

This rule can be “spoken” as:

2 33 2 5d

x x xdx

d(u∙v) = v ∙ du + u ∙ dv

= (2x3 + 5x) (2x) + (x2 + 3) (6x2 + 5)

= (4x4 + 10x2) + (6x4 + 23x2 + 15)

= 10x4 + 33x2 + 15

u = x2 + 3v = 2x3 + 5x

du/dx = dv/dx =

2x + 02∙3x2 + 5

quotient rule:

2

du dvv ud u dx dx

dx v v

or 2

u v du u dvdv v

3

2

2 5

3

d x x

dx x

2 3x

u = 2x3 + 5x

v = x2 + 3

du =

dv =

22 3 5x

2 0x 2 23 6 5x x

2 2 33 6 5 2 5x x x x

2 2 33 6 5 2 5 2x x x x x

2 2 3

22

3 6 5 2 5 2

3

x x x x x

x

Higher Order Derivatives:

dyy

dx is the first derivative of y with respect to x.

2

2

y dy

d

d d d y

dx dx dxxy

is the second derivative

(“y double prime”), thederivative of the first derivative.

3

3

d d y

dy

x

y

x d

is the third derivative

We will see later what these higher-order derivatives might mean in “the real world!”

(“y triple prime”), the derivative of the second derivative.