(d^2)/(dx^2)((y)(x^2+y^2))

解

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

x^{2} \frac{d ^{2} y}{d x ^{2}} + 4 x \frac{d y}{d x} + 2 y + 6 y \frac{d y}{d x}^{2} + 3 y^{2} \frac{d ^{2} y}{d x ^{2}}

解答ステップ

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

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

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

\frac{d}{d x} (x^{2} \frac{d y}{d x} + 2 x y + 3 y^{2} \frac{d y}{d x}) = x^{2} \frac{d ^{2} y}{d x ^{2}} + 4 x \frac{d y}{d x} + 2 y + 6 y \frac{d y}{d x}^{2} + 3 y^{2} \frac{d ^{2} y}{d x ^{2}}

= x^{2} \frac{d ^{2} y}{d x ^{2}} + 4 x \frac{d y}{d x} + 2 y + 6 y \frac{d y}{d x}^{2} + 3 y^{2} \frac{d ^{2} y}{d x ^{2}}