NEU

4. =
=
=
=
=
x
x⋅( −2⋅√2x+1)
√2x +1
x
−2⋅√2x+1
√2x +1
x 4 | x kürzen
x 3 | Subtrahend mit ⋅ √2x+1
√2x+1 erweitern
x
+1
−2⋅√2x+1⋅√2x
√2x +1 √2x +1
x 3
x 2⋅(2x +1)
−
√2x +1 √2x +1
x 3
x−4x−2
√2x+1
x 3
= −3x−2
√2x +1 : x3
= −3x−2 ⋅ 1 −3x−2
√2x +1 x
3=
x 3 ⋅√2x +1
x⋅ln(2x +1)
c) h(x)=
e x
Quotientenregel
1. u(x)=x⋅ln(2x+1) v( x)=e x
2. u'(x)=? v '( x)=e x
Nebenrechnung für u'(x)=(x⋅ln(2x+1))' → Produktregel
1. u(x)=x v( x)=ln(2x+1)
2. u'(x)=1 v '( x)= 1
2x +1 ⋅2= 2
2x+1
3. (x⋅ln(2x+1))'=1⋅ln(2x+1)+x⋅ 2
2x +1
4. =ln(2x+1)+ 2x
2x +1
2x
(ln(2x +1)+
2x +1
3. h '(x)= )⋅ex −(x⋅ln(2x +1))⋅e x
(e x ) 2
4. = ex ⋅(ln(2x+1)+ 2x −x⋅ln(2x +1))
2x+1
| e x kürzen
(e x ) 2
2x
ln(2x+1)+ −x⋅ln( 2x+1)
2x+1
=
e x
2x
ln(2x+1)−x⋅ln(2x +1)+
2x+1
=
e x
=(ln(2x+1)−x⋅ln(2x +1)+ 2x ): ex
2x+1
=(ln(2x+1)⋅(1−x)+ 2x
2x+1 ):ex
= ln(2x+1)⋅(1−x) 2x
+
e x (2x+1)⋅e x