(\partial)/(\partial y)((xy+3x^2)/(1+y))

Lösung

\frac{\partial}{\partial y} (\frac{x y + 3 x ^{2}}{1 + y})

\frac{x - 3 x ^{2}}{( 1 + y ) ^{2}}

Schritte zur Lösung

\frac{\partial}{\partial y} (\frac{x y + 3 x ^{2}}{1 + y})

Behandle

x

als Konstante

Wende die Quotientenregel an:

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

= \frac{\frac{\partial}{\partial y} ( x y + 3 x ^{2} ) ( 1 + y ) - \frac{\partial}{\partial y} ( 1 + y ) ( x y + 3 x ^{2} )}{( 1 + y ) ^{2}}

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

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

= \frac{x ( 1 + y ) - 1 \cdot ( x y + 3 x ^{2} )}{( 1 + y ) ^{2}}

Vereinfache

\frac{x ( 1 + y ) - 1 \cdot ( x y + 3 x ^{2} )}{( 1 + y ) ^{2}} : \frac{x - 3 x ^{2}}{( 1 + y ) ^{2}}

= \frac{x - 3 x ^{2}}{( 1 + y ) ^{2}}

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