derivative of 2/(sqrt(x^3+1))

Lösung

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

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

Schritte zur Lösung

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

Entferne die Konstante:

(a \cdot f)^{'} = a \cdot f^{'}

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

Wende Exponentenregel an:

\frac{1}{a} = a^{- 1}

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

Wende die Kettenregel an:

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

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

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

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

Vereinfache

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

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

y - x y^{^{'}} = a (1 + x^{2} y^{^{'}})

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