derivative of x^{4/5}(x-6^2)

Lösung

\frac{d}{d x} (x^{\frac{4}{5}} (x - 6)^{2})

\frac{14 x ^{2} - 108 x + 144}{5 x ^{\frac{1}{5}}}

Schritte zur Lösung

\frac{d}{d x} (x^{\frac{4}{5}} (x - 6)^{2})

Wende die Produktregel an:

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

f = x^{\frac{4}{5}}, g = (x - 6)^{2}

= \frac{d}{d x} (x^{\frac{4}{5}}) (x - 6)^{2} + \frac{d}{d x} ((x - 6)^{2}) x^{\frac{4}{5}}

\frac{d}{d x} (x^{\frac{4}{5}}) = \frac{4}{5 x ^{\frac{1}{5}}}

\frac{d}{d x} ((x - 6)^{2}) = 2 (x - 6)

= \frac{4}{5 x ^{\frac{1}{5}}} (x - 6)^{2} + 2 (x - 6) x^{\frac{4}{5}}

Vereinfache

\frac{4}{5 x ^{\frac{1}{5}}} (x - 6)^{2} + 2 (x - 6) x^{\frac{4}{5}} : \frac{14 x ^{2} - 108 x + 144}{5 x ^{\frac{1}{5}}}

= \frac{14 x ^{2} - 108 x + 144}{5 x ^{\frac{1}{5}}}

x \to 0 lim (\frac{x}{- 2 x ^{4} - 14 x})

x \to - 2 lim (\frac{x ^{2}}{x ^{2} - 4})

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