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

Lösung

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

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

Schritte zur Lösung

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

Behandle

x

als Konstante

Entferne die Konstante:

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

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

Vereinfache

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

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

\frac{d}{d x} (arcsin (x + lo g_{4} (x)))

d er i v a t i v e 2 t + \frac{4}{3} t^{\frac{3}{2}} + \frac{t ^{2}}{2}

\int_{0}^{2} \frac{1}{1 + x} d x

\int_{- 1}^{1} e^{u + 1} d u

x \to 2 lim (\frac{x ^{2} + 1}{x - 2})