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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Behandle

x

als Konstante

Entferne die Konstante:

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

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

Wende die Quotientenregel an:

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

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

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

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

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

Vereinfache

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

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

Beliebte Beispiele

derivative (x^2)/(8+sqrt(x))

d er i v a t i v e \frac{x ^{2}}{8 + x}

integral of (cos(x))/(sqrt(sin(x)))

\int \frac{cos ( x )}{sin ( x )} d x

sum from n=0 to infinity of 1/n-1/(n+2)

n = 0 \sum \infty \frac{1}{n} - \frac{1}{n + 2}

y^{^{'}} = - 1.5 y

integral of cos^4(x/2)

\int cos^{4} (\frac{x}{2}) d x