(d^2)/(dx^2)((y)(-x^2y^3+4xy^2+12xy))

Lösung

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

- 12 x^{2} y^{2} \frac{d y}{d x}^{2} - 4 x^{2} y^{3} \frac{d ^{2} y}{d x ^{2}} - 16 x y^{3} \frac{d y}{d x} + 24 x y \frac{d y}{d x}^{2} + 12 x y^{2} \frac{d ^{2} y}{d x ^{2}} + 24 x \frac{d y}{d x}^{2} + 24 x y \frac{d ^{2} y}{d x ^{2}} + 24 y^{2} \frac{d y}{d x} + 48 y \frac{d y}{d x} - 2 y^{4}

Schritte zur Lösung

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

\frac{d}{d x} (y (- x^{2} y^{3} + 4 x y^{2} + 12 x y)) = - 4 x^{2} y^{3} \frac{d y}{d x} + 12 x y^{2} \frac{d y}{d x} + 24 x y \frac{d y}{d x} - 2 x y^{4} + 4 y^{3} + 12 y^{2}

= \frac{d}{d x} (- 4 x^{2} y^{3} \frac{d y}{d x} + 12 x y^{2} \frac{d y}{d x} + 24 x y \frac{d y}{d x} - 2 x y^{4} + 4 y^{3} + 12 y^{2})

\frac{d}{d x} (- 4 x^{2} y^{3} \frac{d y}{d x} + 12 x y^{2} \frac{d y}{d x} + 24 x y \frac{d y}{d x} - 2 x y^{4} + 4 y^{3} + 12 y^{2}) = - 12 x^{2} y^{2} \frac{d y}{d x}^{2} - 4 x^{2} y^{3} \frac{d ^{2} y}{d x ^{2}} - 16 x y^{3} \frac{d y}{d x} + 24 x y \frac{d y}{d x}^{2} + 12 x y^{2} \frac{d ^{2} y}{d x ^{2}} + 24 x \frac{d y}{d x}^{2} + 24 x y \frac{d ^{2} y}{d x ^{2}} + 24 y^{2} \frac{d y}{d x} + 48 y \frac{d y}{d x} - 2 y^{4}

= - 12 x^{2} y^{2} \frac{d y}{d x}^{2} - 4 x^{2} y^{3} \frac{d ^{2} y}{d x ^{2}} - 16 x y^{3} \frac{d y}{d x} + 24 x y \frac{d y}{d x}^{2} + 12 x y^{2} \frac{d ^{2} y}{d x ^{2}} + 24 x \frac{d y}{d x}^{2} + 24 x y \frac{d ^{2} y}{d x ^{2}} + 24 y^{2} \frac{d y}{d x} + 48 y \frac{d y}{d x} - 2 y^{4}

s im pl i f y (3 + x 2)^{2}

e x p an d - 34 (232 + 35)

e x p an d 5^{2}

s im pl i f y (3 y - 2)^{2}

e x p an d \frac{( s + 1 ) ^{2}}{s ^{2} - s - 6}