# Mathematrix: Aufgabenbeispiele/ Ableitung von Potenzfunktionen

 i) Berechnen Sie die Ableitung der folgenden Funktionen. A)${\displaystyle \ b(v)=7\ v^{5}\quad }$ B)${\displaystyle \ x(y)=y^{9}\quad }$ C)${\displaystyle \ f(x)={\frac {5}{12}}x^{12}}$ D)${\displaystyle \ a(c)=b\ c^{d}}$ E)${\displaystyle \ v(t)={\sqrt[{4}]{t^{3}}}\quad }$ F)${\displaystyle \ f(x)={\frac {5}{x^{5}}}\quad }$ G)${\displaystyle \ V(h)={\sqrt[{4}]{\frac {1}{h^{5}}}}\quad }$ H)${\displaystyle \ H(x)={\sqrt[{3}]{\frac {27}{x^{7}}}}\quad }$ I)${\displaystyle \ m(n)={\sqrt[{5}]{\frac {3125}{n^{15}}}}\ }$. ii) Berechnen Sie den Wert der Funktion und der Ableitung . an der Stelle 2 bei Aufgaben A, B, C, D, G und I

i)

A) ${\displaystyle \ b(v)=7\ v^{5}\ \Rightarrow \ b'(v)=35\ v^{4}\ \quad }$B) ${\displaystyle \ x(y)=y^{9}\ \Rightarrow \ x'(y)=9\ y^{8}\ \quad }$

C) ${\displaystyle \ f(x)={\frac {5}{12}}x^{12}\ \Rightarrow \ f'(x)={5}x^{11}\ \quad }$D) ${\displaystyle \ a(c)=b\ c^{d}\ \Rightarrow \ a'(c)=b\ d\ c^{d-1}}$

E) ${\displaystyle \ v(t)={\sqrt[{4}]{t^{3}}}\ =t^{\frac {3}{4}}\ \ \Rightarrow \ v'(t)={\frac {3}{4}}t^{{\frac {3}{4}}-1}\ ={\frac {3}{4}}t^{-{\frac {1}{4}}}\left(={\frac {3}{4\ {\sqrt[{4}]{t}}}}\right)\quad }$

F) ${\displaystyle \ f(x)={\frac {5}{x^{5}}}=5\ x^{-5}\ \ \Rightarrow \ f'(x)=5\cdot (-5)\ x^{-5-1}\ =-25\ x^{-6}\quad }$

G) ${\displaystyle \ V(h)={\sqrt[{4}]{\frac {1}{h^{5}}}}\ ={\sqrt[{4}]{h^{-5}}}\ =h^{-{5 \over 4}}\ \Rightarrow \ V'(h)=-{5 \over 4}h^{-{\frac {5}{4}}-1}\ =-{5 \over 4}h^{-{\frac {9}{4}}}\left(=-{\frac {5}{4\ {\sqrt[{4}]{h^{9}}}}}\right)}$

H) ${\displaystyle \ H(x)={\sqrt[{3}]{\frac {27}{x^{7}}}}\ ={\sqrt[{3}]{27}}{\sqrt[{3}]{x^{-7}}}=3\ x^{-{\frac {7}{3}}}\ \Rightarrow \ }$

${\displaystyle \qquad \qquad H'(x)=3\cdot \left(-{\frac {7}{3}}\right)x^{-{\frac {7}{3}}-1}=-7\ x^{-{\frac {10}{3}}}\left(=-{\frac {7}{\sqrt[{3}]{x^{10}\ }}}\right)}$

I) ${\displaystyle \ m(n)={\sqrt[{5}]{\frac {3125}{n^{15}}}}\ ={\sqrt[{5}]{3125}}\ {\sqrt[{5}]{n^{-15}}}=5\ n^{-{\frac {15}{5}}}=5\ n^{-3}\ \Rightarrow \ }$

${\displaystyle \qquad \qquad m'(n)=5\cdot \left(-3\right)n^{-3-1}=-15\ n^{-4}}$.

ii)

A) ${\displaystyle \ b(2)=7\cdot 2^{5}=224\ \ \ b'(2)=35\cdot 2^{4}=560\ \quad }$B) ${\displaystyle \ x(2)=2^{9}=512\ \ \ x'(2)=9\cdot 2^{8}=2304\ \quad }$

C) ${\displaystyle \ f(2)={\frac {5}{12}}\cdot 2^{12}=1706{,}{\dot {6}}\ \ \ f'(x)={\frac {5}{2}}^{11}=20480\ \quad }$D) ${\displaystyle \ a(2)=b\cdot 2^{d}\ \ \ a'(2)=b\cdot d\cdot 2^{d-1}}$

G) ${\displaystyle \ V(2)=2^{-{5 \over 4}}\approx 0{,}42\ \ \ V'(2)=-{5 \over 4}\cdot 2^{\frac {1}{4}}\approx -1{,}05}$

I) ${\displaystyle \ m(2)=5\cdot 2^{-3}=0{,}625\ \ \ m'(2)=-15\cdot 2^{-4}=-0{,}9375}$