(d^2)/(dx^2)((y)(1/x-2/(y^3)))

Lösung

\frac{d ^{2}}{d x ^{2}} ((y) (\frac{1}{x} - \frac{2}{y ^{3}}))

\frac{4 x ^{3} y \frac{d ^{2} y}{d x ^{2}} - 12 x ^{3} \frac{d y}{d x} ^{2} + x ^{2} y ^{4} \frac{d ^{2} y}{d x ^{2}} - 2 x y ^{4} \frac{d y}{d x} + 2 y ^{5}}{x ^{3} y ^{4}}

Schritte zur Lösung

\frac{d ^{2}}{d x ^{2}} ((y) (\frac{1}{x} - \frac{2}{y ^{3}}))

\frac{d}{d x} (y (\frac{1}{x} - \frac{2}{y ^{3}})) = \frac{\frac{d y}{d x}}{x} + \frac{4 \frac{d y}{d x}}{y ^{3}} - \frac{y}{x ^{2}}

= \frac{d}{d x} (\frac{\frac{d y}{d x}}{x} + \frac{4 \frac{d y}{d x}}{y ^{3}} - \frac{y}{x ^{2}})

\frac{d}{d x} (\frac{\frac{d y}{d x}}{x} + \frac{4 \frac{d y}{d x}}{y ^{3}} - \frac{y}{x ^{2}}) = \frac{4 x ^{3} y \frac{d ^{2} y}{d x ^{2}} - 12 x ^{3} \frac{d y}{d x} ^{2} + x ^{2} y ^{4} \frac{d ^{2} y}{d x ^{2}} - 2 x y ^{4} \frac{d y}{d x} + 2 y ^{5}}{x ^{3} y ^{4}}

= \frac{4 x ^{3} y \frac{d ^{2} y}{d x ^{2}} - 12 x ^{3} \frac{d y}{d x} ^{2} + x ^{2} y ^{4} \frac{d ^{2} y}{d x ^{2}} - 2 x y ^{4} \frac{d y}{d x} + 2 y ^{5}}{x ^{3} y ^{4}}

e x p an d (2 - \frac{1}{3} x)^{2}

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

s im pl i f y (a + ib)^{4}

e x p an d (5 x - x^{3})^{2} 8 + x^{2}

e x p an d \frac{x ^{2}}{( x + 1 ) ^{3}}