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

解

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

x^{2} \frac{d ^{2} y}{d x ^{2}} + 4 x \frac{d y}{d x} - 8 x \frac{d y}{d x}^{2} - 8 x y \frac{d ^{2} y}{d x ^{2}} - 16 y \frac{d y}{d x} + 2 y + 12 y^{2} \frac{d y}{d x}^{2} + 4 y^{3} \frac{d ^{2} y}{d x ^{2}} + 8 \frac{d y}{d x}^{2} + 8 y \frac{d ^{2} y}{d x ^{2}}

解答ステップ

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

\frac{d}{d x} (y (x^{2} - 4 x y + y^{3} + 4 y)) = x^{2} \frac{d y}{d x} - 8 x y \frac{d y}{d x} + 2 x y + 4 y^{3} \frac{d y}{d x} + 8 y \frac{d y}{d x} - 4 y^{2}

= \frac{d}{d x} (x^{2} \frac{d y}{d x} - 8 x y \frac{d y}{d x} + 2 x y + 4 y^{3} \frac{d y}{d x} + 8 y \frac{d y}{d x} - 4 y^{2})

\frac{d}{d x} (x^{2} \frac{d y}{d x} - 8 x y \frac{d y}{d x} + 2 x y + 4 y^{3} \frac{d y}{d x} + 8 y \frac{d y}{d x} - 4 y^{2}) = x^{2} \frac{d ^{2} y}{d x ^{2}} + 4 x \frac{d y}{d x} - 8 x \frac{d y}{d x}^{2} - 8 x y \frac{d ^{2} y}{d x ^{2}} - 16 y \frac{d y}{d x} + 2 y + 12 y^{2} \frac{d y}{d x}^{2} + 4 y^{3} \frac{d ^{2} y}{d x ^{2}} + 8 \frac{d y}{d x}^{2} + 8 y \frac{d ^{2} y}{d x ^{2}}

= x^{2} \frac{d ^{2} y}{d x ^{2}} + 4 x \frac{d y}{d x} - 8 x \frac{d y}{d x}^{2} - 8 x y \frac{d ^{2} y}{d x ^{2}} - 16 y \frac{d y}{d x} + 2 y + 12 y^{2} \frac{d y}{d x}^{2} + 4 y^{3} \frac{d ^{2} y}{d x ^{2}} + 8 \frac{d y}{d x}^{2} + 8 y \frac{d ^{2} y}{d x ^{2}}