d/(dy)((xy)/(x^2+y^2))

Lösung

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

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

Schritte zur Lösung

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

Behandle

x

als Konstante

Entferne die Konstante:

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

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

Wende die Quotientenregel an:

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

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

\frac{d y}{d y} = 1

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

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

Vereinfache

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

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

d er i v a t i v e f (x) = \frac{2 - 5 x}{e ^{x}}

x \to \infty lim (5^{x})

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

\frac{\partial}{\partial x} (x cos (π y))

x \to - \infty lim (e^{- x})