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

Lösung

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

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

Schritte zur Lösung

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

Wende die Quotientenregel an:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

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

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

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

Multipliziere aus

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

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