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

Lösung

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

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

Schritte zur Lösung

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

Vereinfache

\frac{- x y}{x ^{2} + y ^{2}} : - \frac{x y}{x ^{2} + y ^{2}}

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

Behandle

y

als Konstante

Entferne die Konstante:

(a \cdot f)^{'} = a \cdot f^{'}

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

Wende die Quotientenregel an:

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

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

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

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

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

Vereinfache

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

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

x \to 3 + lim (\frac{3 - x}{x ^{2} - 9})

\frac{d ^{2}}{d x ^{2}} f (x) = x^{3} + 3 x^{2} - 2 x + 1

\frac{\partial}{\partial x} (2 x y e^{x z})

ma c l a u r in x

2 y^{^{''}} + 3 y^{^{'}} - y = 0