Potenzregel:
f ( x ) = x n {\displaystyle f(x)=x^n}
f ′ ( x ) = n ⋅ x n − 1 {\displaystyle f'(x)=n\cdot x^{n-1}}
Beispiel: f ( x ) = x 3 {\displaystyle f(x)=x^3 }
f ′ ( x ) = x 2 {\displaystyle f'(x)=x^2 }
Faktorregel:
f ( x ) = a ⋅ g ( x ) {\displaystyle f(x)=a\cdot g(x)}
f ′ ( x ) = a ⋅ g ′ ( x ) {\displaystyle f'(x)=a\cdot g'(x)}
Beispiel: f ( x ) = 2 x 2 {\displaystyle f(x)=2x^2 }
f ′ ( x ) = 4 x {\displaystyle f'(x)=4x }
Summenregel:
f ( x ) = g ( x ) + h ( x ) {\displaystyle f(x)=g(x)+ h(x)}
f ′ ( x ) = g ′ ( x ) + h ′ ( x ) {\displaystyle f'(x)=g'(x)+h'(x)}
Beispiel: f ( x ) = x 2 + x 4 {\displaystyle f(x)=x^2 + x^4 }
f ′ ( x ) = 2 x + 4 x 3 {\displaystyle f'(x)=2x+4x^3 }
Produktregel:
f ( x ) = u ( x ) ⋅ v ( x ) {\displaystyle f(x)=u(x)\cdot v(x)}
f ′ ( x ) = u ′ ( x ) ⋅ v ( x ) + u ( x ) ⋅ v ′ ( x ) {\displaystyle f'(x)=u'(x)\cdot v(x)+u(x) \cdot v'(x)}
Beispiel: f ( x ) = 2 x ⋅ ( x 2 + 3 x ) {\displaystyle f(x)=2x \cdot (x^2+3x) }
f ′ ( x ) = 2 ⋅ ( x 2 + 3 x ) + 2 x ⋅ ( 2 x + 3 ) = 2 x 2 + 6 x + 4 x 2 + 6 x = 6 x 2 + 12 x {\displaystyle f'(x)=2 \cdot (x^2+3x)+2x \cdot (2x+3)= 2x^2+6x+4x^2+6x= 6x^2+12x }
Quotientenregel:
f ( x ) = u ( x ) v ( x ) {\displaystyle f(x)= \frac{u(x)}{v(x)}}
f ( x ) = N ⋅ A Z − Z ⋅ A N N 2 {\displaystyle f(x)= \frac{N \cdot AZ-Z\cdot AN}{N^2}}
Beispiel: f ( x ) = 2 x − 1 x + 2 {\displaystyle f(x)= \frac{2x-1}{x+2} }
f ′ ( x ) = ( x + 2 ) ⋅ 2 − ( 2 x − 1 ) ⋅ 1 ( x + 2 ) 2 = 2 x + 4 − 2 x + 1 ( x + 2 ) 2 = 4 x + 5 ( x + 2 ) 2 {\displaystyle f'(x)=\frac{(x+2)\cdot 2-(2x-1) \cdot 1}{(x+2)^2} = \frac{2x+4-2x+1}{(x+2)^2} = \frac{4x+5}{(x+2)^2} }
