(\partial)/(\partial x)((x-y)/(x^2+y^2))

Lösung

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

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

Schritte zur Lösung

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

Behandle

y

als Konstante

Wende die Quotientenregel an:

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

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

\frac{\partial}{\partial x} (x - y) = 1

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

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

Vereinfache

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

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

\int \frac{1 - x}{x ^{2} + 3 x + 2} d x

\frac{d}{d x} (x e^{x} sin (2 x))

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

\int (cos (θ) - 2 sin (θ))^{2} d θ

(x - y + 2) d x - (x - y) d y = 0