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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Behandle

x

als Konstante

Entferne die Konstante:

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

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

Wende die Produktregel an:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = y, g = x^{2} + y^{2}

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

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

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

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

Vereinfache

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

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

Beliebte Beispiele

integral of xsqrt(3x^2+4)

\int x 3 x^{2} + 4 d x

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\int \frac{5 x}{x ^{2} + 1} d x

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n = 1 \sum \infty \frac{1}{n 3 ^{n}}

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\int sin^{3} (15 x) d x