(d^3)/(dx^3)((y)(2x^7-x^4-3x^3))

解

\frac{d ^{3}}{d x ^{3}} ((y) (2 x^{7} - x^{4} - 3 x^{3}))

2 x^{7} \frac{d ^{3} y}{d x ^{3}} + 42 x^{6} \frac{d ^{2} y}{d x ^{2}} + 252 x^{5} \frac{d y}{d x} - x^{4} \frac{d ^{3} y}{d x ^{3}} + 420 x^{4} y - 3 x^{3} \frac{d ^{3} y}{d x ^{3}} - 12 x^{3} \frac{d ^{2} y}{d x ^{2}} - 27 x^{2} \frac{d ^{2} y}{d x ^{2}} - 36 x^{2} \frac{d y}{d x} - 54 x \frac{d y}{d x} - 24 x y - 18 y

解答ステップ

\frac{d ^{3}}{d x ^{3}} ((y) (2 x^{7} - x^{4} - 3 x^{3}))

\frac{d}{d x} (y (2 x^{7} - x^{4} - 3 x^{3})) = \frac{d y}{d x} (2 x^{7} - x^{4} - 3 x^{3}) + (14 x^{6} - 4 x^{3} - 9 x^{2}) y

= \frac{d ^{2}}{d x ^{2}} (\frac{d y}{d x} (2 x^{7} - x^{4} - 3 x^{3}) + (14 x^{6} - 4 x^{3} - 9 x^{2}) y)

\frac{d}{d x} (\frac{d y}{d x} (2 x^{7} - x^{4} - 3 x^{3}) + (14 x^{6} - 4 x^{3} - 9 x^{2}) y) = \frac{d ^{2} y}{d x ^{2}} (2 x^{7} - x^{4} - 3 x^{3}) + 2 (14 x^{6} - 4 x^{3} - 9 x^{2}) \frac{d y}{d x} + (84 x^{5} - 12 x^{2} - 18 x) y

= \frac{d}{d x} (\frac{d ^{2} y}{d x ^{2}} (2 x^{7} - x^{4} - 3 x^{3}) + 2 (14 x^{6} - 4 x^{3} - 9 x^{2}) \frac{d y}{d x} + (84 x^{5} - 12 x^{2} - 18 x) y)

\frac{d}{d x} (\frac{d ^{2} y}{d x ^{2}} (2 x^{7} - x^{4} - 3 x^{3}) + 2 (14 x^{6} - 4 x^{3} - 9 x^{2}) \frac{d y}{d x} + (84 x^{5} - 12 x^{2} - 18 x) y) = 2 x^{7} \frac{d ^{3} y}{d x ^{3}} + 42 x^{6} \frac{d ^{2} y}{d x ^{2}} + 252 x^{5} \frac{d y}{d x} - x^{4} \frac{d ^{3} y}{d x ^{3}} + 420 x^{4} y - 3 x^{3} \frac{d ^{3} y}{d x ^{3}} - 12 x^{3} \frac{d ^{2} y}{d x ^{2}} - 27 x^{2} \frac{d ^{2} y}{d x ^{2}} - 36 x^{2} \frac{d y}{d x} - 54 x \frac{d y}{d x} - 24 x y - 18 y

= 2 x^{7} \frac{d ^{3} y}{d x ^{3}} + 42 x^{6} \frac{d ^{2} y}{d x ^{2}} + 252 x^{5} \frac{d y}{d x} - x^{4} \frac{d ^{3} y}{d x ^{3}} + 420 x^{4} y - 3 x^{3} \frac{d ^{3} y}{d x ^{3}} - 12 x^{3} \frac{d ^{2} y}{d x ^{2}} - 27 x^{2} \frac{d ^{2} y}{d x ^{2}} - 36 x^{2} \frac{d y}{d x} - 54 x \frac{d y}{d x} - 24 x y - 18 y