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

Lösung

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

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

Schritte zur Lösung

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

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

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

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

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

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

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

\frac{d}{d x} (c x^{- 4})

n = 1 \sum \infty 3 (5)^{- n}

x \to 0 lim (9 x - 8 x)

\int_{0}^{\frac{1}{4}} arcsin (2 x) d x

\frac{d}{d x} (\frac{5 x}{x ^{2} - 1})