derivative of x^{1/3}(x^2-4)

Lösung

\frac{d}{d x} (x^{\frac{1}{3}} (x^{2} - 4))

\frac{7 x ^{2} - 4}{3 x ^{\frac{2}{3}}}

Schritte zur Lösung

\frac{d}{d x} (x^{\frac{1}{3}} (x^{2} - 4))

Wende die Produktregel an:

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

f = x^{\frac{1}{3}}, g = x^{2} - 4

= \frac{d}{d x} (x^{\frac{1}{3}}) (x^{2} - 4) + \frac{d}{d x} (x^{2} - 4) x^{\frac{1}{3}}

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

\frac{d}{d x} (x^{2} - 4) = 2 x

= \frac{1}{3 x ^{\frac{2}{3}}} (x^{2} - 4) + 2 x x^{\frac{1}{3}}

Vereinfache

\frac{1}{3 x ^{\frac{2}{3}}} (x^{2} - 4) + 2 x x^{\frac{1}{3}} : \frac{7 x ^{2} - 4}{3 x ^{\frac{2}{3}}}

= \frac{7 x ^{2} - 4}{3 x ^{\frac{2}{3}}}

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