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# MathGymOS/ Analysis/ Differentialrechnung/ Ableitung ganzrationaler Funktionen/ Übungsaufgaben/ Lösungen

Hier sind die Lösungen der Übungsaufgaben zum Ableiten.

## Allgemeines Ableiten

1. ${\displaystyle f(x)=x^{3};\quad f'(x)=3x^{2}\ }$
2. ${\displaystyle f(x)=x^{7};\quad f'(x)=7x^{6}\ }$
3. ${\displaystyle f(x)=x^{1 \over 4};\quad f'(x)={1 \over 4}x^{-{3 \over 4}}\ }$
4. ${\displaystyle f(x)={\sqrt {x}};\quad f'(x)={1 \over 2}x^{-{1 \over 2}}\ }$
5. ${\displaystyle f(x)=x^{-4};\quad f'(x)=-4x^{-5}\ }$
6. ${\displaystyle f(x)=2^{3};\quad f'(x)=-6^{2}\ }$

## Faktorenregel

1. ${\displaystyle f(x)=2x^{4};\quad f'(x)=8x^{3}\ }$
2. ${\displaystyle f(x)=-3x^{3};\quad f'(x)=-9x^{2}\ }$
3. ${\displaystyle f(x)=-x^{1 \over 6};\quad f'(x)=-{1 \over 6}x^{-{5 \over 6}}\ }$
4. ${\displaystyle f(x)={3 \over 4}x^{2};\quad f'(x)={3 \over 2}x\ }$
5. ${\displaystyle f(x)=x;\quad f'(x)=1\ }$
6. ${\displaystyle f(x)=5{\sqrt {x}};\quad f'(x)={5 \over 2}x^{-{1 \over 2}}\ }$
7. ${\displaystyle f(x)=-4;\quad f'(x)=0\ }$

## Summenregel

1. ${\displaystyle f(x)=x^{4}+x^{2};\quad f'(x)=4x^{3}+2x\ }$
2. ${\displaystyle f(x)=x^{3}-x^{1 \over 7};\quad f'(x)=3x^{2}-{1 \over 7}x^{-{6 \over 7}}\ }$
3. ${\displaystyle f(x)=x^{2}+6;\quad f'(x)=2x\ }$
4. ${\displaystyle f(x)=3+4;\quad f'(x)=0\ }$

## Weitere

1. ${\displaystyle f(x)=x^{3}+x^{2};\quad f'(x)=3x^{2}+2x;\quad f''(x)=6x+2\ }$
2. ${\displaystyle f(x)=2x^{4}-4x;\quad f'(x)=8x^{3}-4;\quad f''(x)=24x^{2}\ }$
3. ${\displaystyle f(x)=-5x^{3}+x-6;\quad f'(x)=-15x^{2}+1;\quad f''(x)=-30x\ }$
4. ${\displaystyle f(x)={2 \over 4}x^{4}+x^{2};\quad f'(x)=2x^{3}+2x;\quad f''(x)=6x^{2}+2\ }$
5. ${\displaystyle f(x)=-{3 \over 7}x^{-4}+7x^{2}-{2 \over 3};\quad f'(x)={12 \over 7}x^{-5}+14x;\quad f''(x)=-{60 \over 7}x^{-6}+14\ }$
6. ${\displaystyle f(x)=2x^{3}+1{1 \over 2}x^{7};\quad f'(x)=6x^{2}+{21 \over 2}x^{6};\quad f''(x)=12x+63x^{5}\ }$
7. ${\displaystyle f(x)=0.2x^{2 \over 5};\quad f'(x)={2 \over 25}x^{-{3 \over 5}};\quad f''(x)=-{6 \over 125}x^{-{8 \over 5}}\ }$
8. ${\displaystyle f(x)=x^{5 \over 7}+x^{-3};\quad f'(x)={5 \over 7}x^{-{2 \over 7}}-3x^{-4};\quad f''(x)=-{10 \over 49}x^{-{9 \over 7}}+12x^{-5}\ }$
9. ${\displaystyle f(x)=x+6^{4};\quad f'(x)=1;\quad f''(x)=0\ }$
10. ${\displaystyle f(x)=-{1 \over 6}x^{5}+4x^{4}-{6 \over 7}x^{-2}+2x^{1 \over 3}-x+{23 \over 56};\quad f'(x)=-{5 \over 6}x^{4}+16x^{3}+{12 \over 7}x^{-3}+{2 \over 3}x^{-{2 \over 3}}-1;\quad \ }$

${\displaystyle f''(x)=-{10 \over 3}x^{3}+48x^{2}-{36 \over 7}x^{-4}-{4 \over 9}x^{-{5 \over 3}}\ }$