Analysis
rates of change
-
absolute change
:
f(b) -
f(a)
----
absolute
change
----
o
a D
-
relative (percentage) change :
f(b)
-
f(a)
f(a)
------
f(b)
-...
f(b) -
f(a)
100
percentage change
>
f(a)
-
f(a)
--
....
a b
difference quotient (average of change/mittlern Änderungsrate)
f(b)-l
:
-
rale
change over an interval !
sequante
G
average change
81
a b
-
differential quotient (instantaneous rak of change/momentane Änderungsrate) :
f(x) =Cimf
change at one specific point !
tangent
instan
,
, Kinematics
displacement s(t)
differentiales velocity v(t) ~
integrate
sacceleration alt)
H) It c
displacement
Sv(t) It < total distance travelled
Rules of differentiation
x" f(x) +"
1
f(x)
=
-
n
-
.
=
a - -
a
k g(x) f(x) k
g'(x)(k constant)
f(x)
-
.
2 = -
= -
...
3 .
f(x) =
g(x) =
h(x) +
f(x)
=
g(x) = h'(x)
Chain Rule : Quotient Rule :
y f(g(x) u(x)
g(x)"
=
y
=
f(x) =
v(x)
n -
1 y' =
g'(x) -
f'(x)
y
=
n
g(x)
-
g(x) f(x) u'(x) v(x) u(x) v'(x)
-
=
-
- .
2
.
et cos(x2 b) -
v(x)
ex .
(Sin(x)5 -
f 3 ex x + 6
(sin(x)"
.
u(r) u'(r) v(x) v'(x)
5 .
(cos(t)) ·
-
sin (x" 3) - .
2x 2x + 1
x2 + 62x2x + 12
f(x) g(x
(2x) (2x 1) (2 + 6)
.
+
2
-
-
.
[2x 1]2 +
Product Rule :
f(x) =
u(x) .
v(x)
f'(x) =
u((x) v(x) .
+ v((x) u(x) -
(x 4)
5
f =
(x 4)af) 5x4 + 2
(x 4)
.
-
= +
.
= -
ex
.
x
-
.
u(r) u'(r) v(x) v'(x)
x55"( 4)22 (x 4) -
.
-
rates of change
-
absolute change
:
f(b) -
f(a)
----
absolute
change
----
o
a D
-
relative (percentage) change :
f(b)
-
f(a)
f(a)
------
f(b)
-...
f(b) -
f(a)
100
percentage change
>
f(a)
-
f(a)
--
....
a b
difference quotient (average of change/mittlern Änderungsrate)
f(b)-l
:
-
rale
change over an interval !
sequante
G
average change
81
a b
-
differential quotient (instantaneous rak of change/momentane Änderungsrate) :
f(x) =Cimf
change at one specific point !
tangent
instan
,
, Kinematics
displacement s(t)
differentiales velocity v(t) ~
integrate
sacceleration alt)
H) It c
displacement
Sv(t) It < total distance travelled
Rules of differentiation
x" f(x) +"
1
f(x)
=
-
n
-
.
=
a - -
a
k g(x) f(x) k
g'(x)(k constant)
f(x)
-
.
2 = -
= -
...
3 .
f(x) =
g(x) =
h(x) +
f(x)
=
g(x) = h'(x)
Chain Rule : Quotient Rule :
y f(g(x) u(x)
g(x)"
=
y
=
f(x) =
v(x)
n -
1 y' =
g'(x) -
f'(x)
y
=
n
g(x)
-
g(x) f(x) u'(x) v(x) u(x) v'(x)
-
=
-
- .
2
.
et cos(x2 b) -
v(x)
ex .
(Sin(x)5 -
f 3 ex x + 6
(sin(x)"
.
u(r) u'(r) v(x) v'(x)
5 .
(cos(t)) ·
-
sin (x" 3) - .
2x 2x + 1
x2 + 62x2x + 12
f(x) g(x
(2x) (2x 1) (2 + 6)
.
+
2
-
-
.
[2x 1]2 +
Product Rule :
f(x) =
u(x) .
v(x)
f'(x) =
u((x) v(x) .
+ v((x) u(x) -
(x 4)
5
f =
(x 4)af) 5x4 + 2
(x 4)
.
-
= +
.
= -
ex
.
x
-
.
u(r) u'(r) v(x) v'(x)
x55"( 4)22 (x 4) -
.
-
