derivative of (-9/(sqrt(x^2+81)))

Lösung

\frac{d}{d x} (\frac{- 9}{x ^{2} + 81})

\frac{9 x}{( x ^{2} + 81 ) ^{\frac{3}{2}}}

Schritte zur Lösung

\frac{d}{d x} (\frac{- 9}{x ^{2} + 81})

Vereinfache

\frac{- 9}{x ^{2} + 81} : - \frac{9}{x ^{2} + 81}

= \frac{d}{d x} (- \frac{9}{x ^{2} + 81})

Entferne die Konstante:

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

= - 9 \frac{d}{d x} (\frac{1}{x ^{2} + 81})

Wende Exponentenregel an:

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

= - 9 \frac{d}{d x} ((x^{2} + 81)^{- \frac{1}{2}})

Wende die Kettenregel an:

- \frac{1}{2 ( x ^{2} + 81 ) ^{\frac{3}{2}}} \frac{d}{d x} (x^{2} + 81)

= - \frac{1}{2 ( x ^{2} + 81 ) ^{\frac{3}{2}}} \frac{d}{d x} (x^{2} + 81)

\frac{d}{d x} (x^{2} + 81) = 2 x

= - 9 (- \frac{1}{2 ( x ^{2} + 81 ) ^{\frac{3}{2}}} \cdot 2 x)

Vereinfache

- 9 (- \frac{1}{2 ( x ^{2} + 81 ) ^{\frac{3}{2}}} \cdot 2 x) : \frac{9 x}{( x ^{2} + 81 ) ^{\frac{3}{2}}}

= \frac{9 x}{( x ^{2} + 81 ) ^{\frac{3}{2}}}

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