2.2 Basic Differentiation Rules& Rates of Change
The derivative of a constant function is 0.
That is, if is a real number, then
The Con
st
ant ul
0
R e
c
dc
dx
Quiz
Quiz2
2.2 Basic Differentiation Rules& Rates of Change
Function Derivative
7y 0dy
dx
0f x ' 0f x 3s t ' 0s t
2 , is constanty k k ' 0y
1
If is a rational number, then the function
is differentiable a
The Power
nd
.
Rule
n
n n
n
f x x
dx nx
dx
2.2 Basic Differentiation Rules& Rates of Change
Function Derivative
07y x 0dy
dx
y x ' 1y 3s t t 2' 3s t t
3g x x 2 / 32 / 3
1 1'
3 3g x
x
2
1y
x 2 3
3
22
dy dx x
dx dx x
2.2 Basic Differentiation Rules& Rates of Change
4
3
Find the slope of when
a 1 b 0 c 1
' 4
' 1
' 0
' 1
4
0
4
f x x
x x x
f x x
f
f
f
p. 104
2
tangent
1 1
Find an equation of the tangent line to the graph
of when 2.
' 2 ' 2 4 4
2 4 the tangent goes through 2,4 .
4 4 2
f x x x
f x x f
y
y y m
m
x
f
x x
p. 104
AP ExamMultiple Choice
If is a differentiable function and
is a real number, then is also
differentiable and
The Constant Multip
'
le Rule
.
f
c cf
dcf x cf x
dx
2.2 Basic Differentiation Rules& Rates of Change
Function Derivative
2 /y x 22 /dy
xdx
24
5
tf t 8
'5
f t t
2y x1dy
dx x
3 2
1
2y
x
5 / 3
1
3
dy
dx x
3
2
xy
3'
2y
Function Rewrite Differentiate Simplify
3
5
2y
x 35
2y x
415'
2y x
4
15'
2y
x
3
5
2y
x 35
8y x
415'
8y x
4
15'
8y
x
2
7
3y
x 27
3y x
14'
3y x
2
7
3y
x 263y x ' 126y x
2.2 Basic Differentiation Rules& Rates of Change
The derivative of the sum or difference of two
differentiable functions is differentiable and is the
sum or difference of their derivative
The Sum and Differe
Sum Rule
nce Rules
s.
' '
df x g x f x g x
dxd
' Difference Rule' f x g x f x g xdx
2.2 Basic Differentiation Rules& Rates of Change
Function Derivative
3 4 5f x x x 2' 3 4f x x
4
33 22
xg x x x 3 2' 2 9 2g x x x
2.2 Basic Differentiation Rules& Rates of Change
Derivatives of Sine and Cosine Functio
sin cos
ns
cos sind d
x x x xdx dx
2.2 Basic Differentiation Rules& Rates of Change
Function Derivative
2sinf x x ' 2cosf x xsin
2
xy
1' cos
2y x
2.2 Basic Differentiation Rules& Rates of Change
cosy x x ' 1 siny x
Change in Distance
Change in TimeAverage Velocity
s
t
2
If a billiard ball is dropped from a height of 100 feet, its height
at time is given by
16 100
where is measured in feet and is measured in seconds. Fi
Position Function
s t
s t
s t
nd
the average velocity over each of the following time intervals.
(a) [1,2] (b) [1,1.5] (c) [1,1.1] 2 1 36 84
48 feet per seco1
n1
d2
s ss
t
1.5 1 20
1.5 1 0.40 feet per secon
5d
s ss
t
1.1 1 80.64 84
33.6 feet pe1
r .1 1 0
secon1
d.
s ss
t
Instantaneous Velocity
p. 109
1 2
1
The average velocity between and is the slope
of the secant line, and the instantaneous velocity at
is the slope of the tangent line.
t t
t Avg. Velocity
p. 109
2
At time 0, a diver jumps from a diving board that is 32 feet
above the water. The position of the diver is given by
where is measured in feet and is
16 16
measured in s
32
econ
t
s
s t
t
t t
ds.
(a) When does the diver hit the water?
(b) What is the diver's velocity at impact?
2
2
16 16 32 0
16 2 0
16 2 1 0
2
48 feet per second
' 32 16
' 2
s t t t
t t
t t
s t
t
t
s
HW 2.2/1,3,11,15,17,19,21,24,26,28,29,33,35,37,41,45,46,49
Director: Tim SarkelaProducer: Joey Kurz
Screen Writer: Tim Sarkela
Cameo: Joey KurzTom Cruise
Britney SpearsGeorge W (the dog) Bush
Carl MarxOld Major
Winston SmithRaphael, Michelangelo, and Leonardo
Ken PoynerDarlene SarkelaPatty Granado
The three little pigsDracula
ElvisScorpionSub-zero
Cinematographer: Marin PalmerComposer: Josiah Stocker
Editor: Emily SherryExecutive Producer: Donkey Kong
Key grip: Raul “ricky” MarwahVoice: Rush Limbah
Howard Stern
Characters: Mr. Anderson: Tim Sarkela
Mr. Smith: Joey KurzGizmo: Joey Kurz
Mr. Hoshagowa: Tim Sarkela3rd lady: Joey Kurz
Pregnant man: Tim SarkelaLittle boy: Joey Kurz
Bruce the fun lovin moose: Tim SarkelaSarah: Joey Kurz
Jimmy: Tim SarkelaLassie: Joey KurzButch: Tim Sarkela
Mr. Gibbons: as himself
Special thanks to:Bob’s fish parlor
The Communist partyThe Clinton administration
Nintendo of America
No thanks to:Fred’s fish parlorThe Nazi party
The Bush administrationNintendo of Japan
And a very special thanks to Mr. Gibbons for teaching us these crazy derivatives
Sketch the graph of a function whose
derivative is always negative.
Lesson
3Find the derivative of 2 4.s t t t
2' 3 2s t t
Lesson