derivative of 1/(sqrt(x^2+y^2-25))

Lösung

\frac{d}{d x} (\frac{1}{x ^{2} + y ^{2} - 25})

- \frac{x + y \frac{d y}{d x}}{( x ^{2} + y ^{2} - 25 ) ^{\frac{3}{2}}}

Schritte zur Lösung

\frac{d}{d x} (\frac{1}{x ^{2} + y ^{2} - 25})

Wende Exponentenregel an:

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

= \frac{d}{d x} ((x^{2} + y^{2} - 25)^{- \frac{1}{2}})

Wende die Kettenregel an:

- \frac{1}{2 ( x ^{2} + y ^{2} - 25 ) ^{\frac{3}{2}}} \frac{d}{d x} (x^{2} + y^{2} - 25)

= - \frac{1}{2 ( x ^{2} + y ^{2} - 25 ) ^{\frac{3}{2}}} \frac{d}{d x} (x^{2} + y^{2} - 25)

\frac{d}{d x} (x^{2} + y^{2} - 25) = 2 x + 2 y \frac{d y}{d x}

= - \frac{1}{2 ( x ^{2} + y ^{2} - 25 ) ^{\frac{3}{2}}} (2 x + 2 y \frac{d y}{d x})

Vereinfache

- \frac{1}{2 ( x ^{2} + y ^{2} - 25 ) ^{\frac{3}{2}}} (2 x + 2 y \frac{d y}{d x}) : - \frac{x + y \frac{d y}{d x}}{( x ^{2} + y ^{2} - 25 ) ^{\frac{3}{2}}}

= - \frac{x + y \frac{d y}{d x}}{( x ^{2} + y ^{2} - 25 ) ^{\frac{3}{2}}}

\int e^{t x - x} d x

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