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Beliebt Rechnen >

(\partial)/(\partial y)((x/y)e^{xy})

Lösung

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

Lösung

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

Schritte zur Lösung

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

Behandle

x

als Konstante

Entferne die Konstante:

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

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

Wende die Produktregel an:

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

f = \frac{1}{y}, g = e^{x y}

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

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

\frac{\partial}{\partial y} (e^{x y}) = e^{x y} x

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

Vereinfache

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

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

Beliebte Beispiele

(\partial)/(\partial z)(e^xsin(y))

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

y^{''}-10y^'+25y=t^{-8}e^{5t}

y^{^{''}} - 10 y^{^{'}} + 25 y = t^{- 8} e^{5 t}

integral of 7xe^{5x}

\int 7 x e^{5 x} d x

integral of (2-3x)cos(4x)

\int (2 - 3 x) cos (4 x) d x

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

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