erweitern (2x(-x\frac{dy)/(dx)+y)}{y^3}

Lösung

erweitern

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

- \frac{2 x ^{2} \frac{d y}{d x}}{y ^{3}} + \frac{2 x}{y ^{2}}

Schritte zur Lösung

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

Multipliziere aus

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

= \frac{- 2 x ^{2} \frac{d y}{d x} + 2 x y}{y ^{3}}

Wende Bruchregel an:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{- 2 x ^{2} \frac{d y}{d x} + 2 x y}{y ^{3}} = - \frac{2 x ^{2} \frac{d y}{d x}}{y ^{3}} + \frac{2 x y}{y ^{3}}

= - \frac{2 x ^{2} \frac{d y}{d x}}{y ^{3}} + \frac{2 x y}{y ^{3}}

Streiche

\frac{2 x y}{y ^{3}} : \frac{2 x}{y ^{2}}

= - \frac{2 x ^{2} \frac{d y}{d x}}{y ^{3}} + \frac{2 x}{y ^{2}}

e x p an d \frac{( x + 2 ) ( x - 3 )}{x}

e x p an d \frac{x ( 2 x ^{3} + y )}{6 y ^{2}}

e x p an d \frac{\frac{d}{d x} ( x ^{3} - 2 )}{( 4 x + 3 ) ( 6 x - 2 )}

e x p an d (a + b) a^{3} - a^{2} b + a b^{2}

e x p an d \frac{sin ^{2} ( x ) ( 1 + cos ( x ) )}{1 - cos ( x )}