derivative 1/(sqrt(x^2+a^2))

Lösung

ableitung von

\frac{1}{x ^{2} + a ^{2}}

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

Schritte zur Lösung

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

Wende Exponentenregel an:

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

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

Wende die Kettenregel an:

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

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

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

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

Vereinfache

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

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

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