Date post: | 26-Mar-2015 |
Category: |
Documents |
Upload: | jada-mclain |
View: | 212 times |
Download: | 0 times |
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.