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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Behandle

x

als Konstante

Entferne die Konstante:

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

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

Wende Exponentenregel an:

\frac{1}{a} = a^{- 1}

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

Wende die Kettenregel an:

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

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

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

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

Vereinfache

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

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

Beliebte Beispiele

integral of 9xe^{-x}

\int 9 x e^{- x} d x

integral from 0 to 1 of x^3-3x^2+2x

\int_{0}^{1} x^{3} - 3 x^{2} + 2 x d x

derivative y=x^2-5x+5

d er i v a t i v e y = x^{2} - 5 x + 5

integral of x^2(6-x)

\int x^{2} (6 - x) d x

2y^{''}-y^'-y=6e^{-x/2}

2 y^{^{''}} - y^{^{'}} - y = 6 e^{- \frac{x}{2}}