(d^2)/(dx^2)((y^2)(x^2-2xy+3y^2-20y))

解

\frac{d ^{2}}{d x ^{2}} ((y^{2}) (x^{2} - 2 x y + 3 y^{2} - 20 y))

2 x^{2} \frac{d y}{d x}^{2} + 2 x^{2} y \frac{d ^{2} y}{d x ^{2}} + 8 x y \frac{d y}{d x} - 12 x y \frac{d y}{d x}^{2} - 6 x y^{2} \frac{d ^{2} y}{d x ^{2}} - 12 y^{2} \frac{d y}{d x} + 2 y^{2} + 36 y^{2} \frac{d y}{d x}^{2} + 12 y^{3} \frac{d ^{2} y}{d x ^{2}} - 120 y \frac{d y}{d x}^{2} - 60 y^{2} \frac{d ^{2} y}{d x ^{2}}

解答ステップ

\frac{d ^{2}}{d x ^{2}} ((y^{2}) (x^{2} - 2 x y + 3 y^{2} - 20 y))

\frac{d}{d x} (y^{2} (x^{2} - 2 x y + 3 y^{2} - 20 y)) = 2 x^{2} y \frac{d y}{d x} - 6 x y^{2} \frac{d y}{d x} + 2 x y^{2} + 12 y^{3} \frac{d y}{d x} - 60 y^{2} \frac{d y}{d x} - 2 y^{3}

= \frac{d}{d x} (2 x^{2} y \frac{d y}{d x} - 6 x y^{2} \frac{d y}{d x} + 2 x y^{2} + 12 y^{3} \frac{d y}{d x} - 60 y^{2} \frac{d y}{d x} - 2 y^{3})

\frac{d}{d x} (2 x^{2} y \frac{d y}{d x} - 6 x y^{2} \frac{d y}{d x} + 2 x y^{2} + 12 y^{3} \frac{d y}{d x} - 60 y^{2} \frac{d y}{d x} - 2 y^{3}) = 2 x^{2} \frac{d y}{d x}^{2} + 2 x^{2} y \frac{d ^{2} y}{d x ^{2}} + 8 x y \frac{d y}{d x} - 12 x y \frac{d y}{d x}^{2} - 6 x y^{2} \frac{d ^{2} y}{d x ^{2}} - 12 y^{2} \frac{d y}{d x} + 2 y^{2} + 36 y^{2} \frac{d y}{d x}^{2} + 12 y^{3} \frac{d ^{2} y}{d x ^{2}} - 120 y \frac{d y}{d x}^{2} - 60 y^{2} \frac{d ^{2} y}{d x ^{2}}

= 2 x^{2} \frac{d y}{d x}^{2} + 2 x^{2} y \frac{d ^{2} y}{d x ^{2}} + 8 x y \frac{d y}{d x} - 12 x y \frac{d y}{d x}^{2} - 6 x y^{2} \frac{d ^{2} y}{d x ^{2}} - 12 y^{2} \frac{d y}{d x} + 2 y^{2} + 36 y^{2} \frac{d y}{d x}^{2} + 12 y^{3} \frac{d ^{2} y}{d x ^{2}} - 120 y \frac{d y}{d x}^{2} - 60 y^{2} \frac{d ^{2} y}{d x ^{2}}