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

Lösung

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

\frac{x ^{5} \frac{d y}{d x} + x ^{4} y - 4 x ^{3} y ^{2} \frac{d y}{d x} + 4 x ^{2} y ^{3} - x y ^{4} \frac{d y}{d x} - y ^{5}}{( x ^{2} + y ^{2} ) ^{2}}

Schritte zur Lösung

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

Wende die Quotientenregel an:

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

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

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

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

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

Multipliziere aus

(x^{3} \frac{d y}{d x} + 3 x^{2} y - 3 x y^{2} \frac{d y}{d x} - y^{3}) (x^{2} + y^{2}) - (2 x + 2 y \frac{d y}{d x}) x y (x^{2} - y^{2}) : x^{5} \frac{d y}{d x} + x^{4} y - 4 x^{3} y^{2} \frac{d y}{d x} + 4 x^{2} y^{3} - x y^{4} \frac{d y}{d x} - y^{5}

= \frac{x ^{5} \frac{d y}{d x} + x ^{4} y - 4 x ^{3} y ^{2} \frac{d y}{d x} + 4 x ^{2} y ^{3} - x y ^{4} \frac{d y}{d x} - y ^{5}}{( x ^{2} + y ^{2} ) ^{2}}