derivative of 1/(2ln(x^2+y^2))

Lösung

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

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

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Wende Exponentenregel an:

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

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

Wende die Kettenregel an:

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

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

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

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

Vereinfache

\frac{1}{2} (- \frac{1}{ln ^{2} ( x ^{2} + y ^{2} )} \cdot \frac{2 x + 2 y \frac{d y}{d x}}{x ^{2} + y ^{2}}) : - \frac{x + y \frac{d y}{d x}}{ln ^{2} ( x ^{2} + y ^{2} ) ( x ^{2} + y ^{2} )}

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

d er i v a t i v e \frac{( 3 x - 1 )}{( 2 x + 1 )}

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d er i v a t i v e (\frac{x}{3})^{2}