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

\frac{d}{d x} (arctan (\frac{1}{2} x))

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

\frac{d y}{d x} - \frac{2}{x} y = x^{2} - 1

ma c l a u r in x^{2} cos (x^{5})