Summary
Zusammenfassung Mathe: Ableitungen
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Es ist eine ausführliche Zusammenfassung, die die Themen Differentialgleichungen, Ableitung, Funktionen und Funktionsgleichungen umfasst.
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Zusammenfassung Arbeit Nr.1 25.11.211. Ableitung näherungsweise graphisch bestimmen
Vorgehensweise :
Beispiel
Punkt 0,5×2
'
1. Gegeben : Graph mit Gleichung ,
,
Wert f (a) 1 . flx ) =
, Tangente t PC 212 ) ,
Wer t f- ' (2) bestimmen
2. Tangente im Punkt anlegen 2. Tangente siehe Bild
3. Wer t entspricht Steigung der Tangente 3 .
f. (2) =
Steigung tangente
4. 2 Punkte bestimmen , Yz -
Yi 4 . Punkt 11212 ) Punkt 211,5 , 1)
m =
" -
✗^
"z Yn 1- z
Steigung ausrechnen
-
: m = = =
2
✗ z
-
+
, 1,5 -
z
5. Ergebnis ist gesuchte Steigung ( Ableitung )
5. f (2) 52
'
Ableitungs Graphen :
Wenn der Graph fällt , liegt der Ableitungs Graph unter der
✗ Achse
wenn Graph eine Extremste , ,e hat , na + n.ae#ungsgrapn eine
Nullstelle
2. Rechnerisch Ableiten
Praktische Umformungen :
j =
12 Ableiten von Sin , Cos , 1- an :
✗
^
^ sin
I
-
= ✗
1-
2
- 2 -
Cos COS
✗ = × |
1-
= ×
-
E - sin <
✗
Die Produkt regel Beispiel :
Regel : f. ( x ) = v. ( X ) .
✓ ( x ) t u / ✗ I. ✓
'
(x) fcx ) -
_ F. ( 0511 -
x )
wenn f- ( x ) = vlt ) -
✓ (x ) so gilt : 1 .
.vn ) und ✓ ( × ) bestimmen v (✗ 1=+2 v. 4) = 2x
,
UH ) und ✓(× ) jeweils ableiten ( ) Cos ( A ) '
( x) sin ( 1- × )
v. (x ) ) ✓ (x ) ( ) 2 ✓ x = -
' x ✓ =
f. ( ✗ )
-
✓ (x + ✓ x
. .
= .
3. in f /× )
'
= v. (x ) -
V (x ) tv (x ) .
✓
'
( x ) einsetzen f- ( X )
'
= 2X . COS ( A -
x ) t ✗
2. ( sind
-
-
✗ 1)
lineare Verkettung :
Beispiel :
Regel : f-
'
Ix ) = u
'
(v ( x) ) -
v
'
f (x ) = 15×+2 )
}
}
1. µ (x ) und ✓ ( x ) bestimmen
U (x ) = × ✓
'
(x ) = 3×2
2. ✓ (x ) und v4 ) jeweils ableiten ✓ ( ✗ 1=5×+2 ✓
'
( ✗ 1=5
"
'
f /× ) 5
'
v. ( vie ) ) ( x) =3 (5×+2)
'
3. in = -
v einsetzen f- .
.
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