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

Lösung

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

\frac{- x ^{2} + 6 x y + y ^{2}}{( x ^{2} + y ^{2} ) ^{2}}

Schritte zur Lösung

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

Behandle

y

als Konstante

Wende die Quotientenregel an:

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

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

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

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

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

Vereinfache

\frac{1 \cdot ( x ^{2} + y ^{2} ) - 2 x ( x - 3 y )}{( x ^{2} + y ^{2} ) ^{2}} : \frac{- x ^{2} + 6 x y + y ^{2}}{( x ^{2} + y ^{2} ) ^{2}}

= \frac{- x ^{2} + 6 x y + y ^{2}}{( x ^{2} + y ^{2} ) ^{2}}

\frac{d}{d x} (arctan (\frac{x}{1 + x}))

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\int_{0}^{l n (3)} \frac{4 e ^{x}}{1 + e ^{2 x}} d x