derivative of x(x^2-5^{1/3})

Lösung

\frac{d}{d x} (x (x^{2} - 5)^{\frac{1}{3}})

\frac{5 x ^{2} - 15}{3 ( x ^{2} - 5 ) ^{\frac{2}{3}}}

Schritte zur Lösung

\frac{d}{d x} (x (x^{2} - 5)^{\frac{1}{3}})

Wende die Produktregel an:

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

f = x, g = (x^{2} - 5)^{\frac{1}{3}}

= \frac{d x}{d x} (x^{2} - 5)^{\frac{1}{3}} + \frac{d}{d x} ((x^{2} - 5)^{\frac{1}{3}}) x

\frac{d x}{d x} = 1

\frac{d}{d x} ((x^{2} - 5)^{\frac{1}{3}}) = \frac{2 x}{3 ( x ^{2} - 5 ) ^{\frac{2}{3}}}

= 1 \cdot (x^{2} - 5)^{\frac{1}{3}} + \frac{2 x}{3 ( x ^{2} - 5 ) ^{\frac{2}{3}}} x

Vereinfache

1 \cdot (x^{2} - 5)^{\frac{1}{3}} + \frac{2 x}{3 ( x ^{2} - 5 ) ^{\frac{2}{3}}} x : \frac{5 x ^{2} - 15}{3 ( x ^{2} - 5 ) ^{\frac{2}{3}}}

= \frac{5 x ^{2} - 15}{3 ( x ^{2} - 5 ) ^{\frac{2}{3}}}

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