derivative x^{1/3}(x+3)^{2/3}

Lösung

ableitung von

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

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

Schritte zur Lösung

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

Wende die Produktregel an:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x^{\frac{1}{3}}, g = (x + 3)^{\frac{2}{3}}

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

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

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

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

Vereinfache

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

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

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