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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Behandle

y

als Konstante

Entferne die Konstante:

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

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

Wende Exponentenregel an:

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

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

Wende die Kettenregel an:

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

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

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

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

Vereinfache

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

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

Beliebte Beispiele

integral of sin(x)+y

\int sin (x) + y d x

(\partial)/(\partial y)((2xy^2)/(3+x^2y^2))

\frac{\partial}{\partial y} (\frac{2 x y ^{2}}{3 + x ^{2} y ^{2}})

integral of x^7arctan(x^4)

\int x^{7} arctan (x^{4}) d x

derivative of 4x^2+1/(x^2+3x^4)

\frac{d}{d x} (4 x^{2} + \frac{1}{x ^{2}} + 3 x^{4})

derivative of 8x-5

\frac{d}{d x} (8 x - 5)