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

Lösung

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

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

Schritte zur Lösung

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

Wende die Quotientenregel an:

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

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

\frac{d x}{d x} = 1

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

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

Vereinfache

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

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