(d^3)/(dx^3)((y)(x^4-2x^3+3x-1))

解

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

x^{4} \frac{d ^{3} y}{d x ^{3}} - 2 x^{3} \frac{d ^{3} y}{d x ^{3}} + 12 x^{3} \frac{d ^{2} y}{d x ^{2}} - 18 x^{2} \frac{d ^{2} y}{d x ^{2}} + 36 x^{2} \frac{d y}{d x} + 3 x \frac{d ^{3} y}{d x ^{3}} - 36 x \frac{d y}{d x} + 24 x y - \frac{d ^{3} y}{d x ^{3}} + 9 \frac{d ^{2} y}{d x ^{2}} - 12 y

解答ステップ

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

\frac{d}{d x} (y (x^{4} - 2 x^{3} + 3 x - 1)) = \frac{d y}{d x} (x^{4} - 2 x^{3} + 3 x - 1) + (4 x^{3} - 6 x^{2} + 3) y

= \frac{d ^{2}}{d x ^{2}} (\frac{d y}{d x} (x^{4} - 2 x^{3} + 3 x - 1) + (4 x^{3} - 6 x^{2} + 3) y)

\frac{d}{d x} (\frac{d y}{d x} (x^{4} - 2 x^{3} + 3 x - 1) + (4 x^{3} - 6 x^{2} + 3) y) = \frac{d ^{2} y}{d x ^{2}} (x^{4} - 2 x^{3} + 3 x - 1) + 2 (4 x^{3} - 6 x^{2} + 3) \frac{d y}{d x} + (12 x^{2} - 12 x) y

= \frac{d}{d x} (\frac{d ^{2} y}{d x ^{2}} (x^{4} - 2 x^{3} + 3 x - 1) + 2 (4 x^{3} - 6 x^{2} + 3) \frac{d y}{d x} + (12 x^{2} - 12 x) y)

\frac{d}{d x} (\frac{d ^{2} y}{d x ^{2}} (x^{4} - 2 x^{3} + 3 x - 1) + 2 (4 x^{3} - 6 x^{2} + 3) \frac{d y}{d x} + (12 x^{2} - 12 x) y) = x^{4} \frac{d ^{3} y}{d x ^{3}} - 2 x^{3} \frac{d ^{3} y}{d x ^{3}} + 12 x^{3} \frac{d ^{2} y}{d x ^{2}} - 18 x^{2} \frac{d ^{2} y}{d x ^{2}} + 36 x^{2} \frac{d y}{d x} + 3 x \frac{d ^{3} y}{d x ^{3}} - 36 x \frac{d y}{d x} + 24 x y - \frac{d ^{3} y}{d x ^{3}} + 9 \frac{d ^{2} y}{d x ^{2}} - 12 y

= x^{4} \frac{d ^{3} y}{d x ^{3}} - 2 x^{3} \frac{d ^{3} y}{d x ^{3}} + 12 x^{3} \frac{d ^{2} y}{d x ^{2}} - 18 x^{2} \frac{d ^{2} y}{d x ^{2}} + 36 x^{2} \frac{d y}{d x} + 3 x \frac{d ^{3} y}{d x ^{3}} - 36 x \frac{d y}{d x} + 24 x y - \frac{d ^{3} y}{d x ^{3}} + 9 \frac{d ^{2} y}{d x ^{2}} - 12 y