(\partial)/(\partial y)((x^3+(y/2))^{2/3})

Lösung

\frac{\partial}{\partial y} ((x^{3} + (\frac{y}{2}))^{\frac{2}{3}})

\frac{2 ^{\frac{1}{3}}}{3 ( 2 x ^{3} + y ) ^{\frac{1}{3}}}

Schritte zur Lösung

\frac{\partial}{\partial y} ((x^{3} + (\frac{y}{2}))^{\frac{2}{3}})

Vereinfache

(x^{3} + (\frac{y}{2}))^{\frac{2}{3}} : (\frac{2 x ^{3} + y}{2})^{\frac{2}{3}}

= \frac{\partial}{\partial y} ((\frac{2 x ^{3} + y}{2})^{\frac{2}{3}})

Behandle

x

als Konstante

Wende die Kettenregel an:

\frac{2}{3 ( \frac{2 x ^{3} + y}{2} ) ^{\frac{1}{3}}} \frac{\partial}{\partial y} (\frac{2 x ^{3} + y}{2})

= \frac{2}{3 ( \frac{2 x ^{3} + y}{2} ) ^{\frac{1}{3}}} \frac{\partial}{\partial y} (\frac{2 x ^{3} + y}{2})

\frac{\partial}{\partial y} (\frac{2 x ^{3} + y}{2}) = \frac{1}{2}

= \frac{2}{3 ( \frac{2 x ^{3} + y}{2} ) ^{\frac{1}{3}}} \cdot \frac{1}{2}

Vereinfache

\frac{2}{3 ( \frac{2 x ^{3} + y}{2} ) ^{\frac{1}{3}}} \cdot \frac{1}{2} : \frac{2 ^{\frac{1}{3}}}{3 ( 2 x ^{3} + y ) ^{\frac{1}{3}}}

= \frac{2 ^{\frac{1}{3}}}{3 ( 2 x ^{3} + y ) ^{\frac{1}{3}}}

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