(d^3)/(dx^3)((y)(5x^3+3x^2-2x+9))

解

\frac{d ^{3}}{d x ^{3}} ((y) (5 x^{3} + 3 x^{2} - 2 x + 9))

5 x^{3} \frac{d ^{3} y}{d x ^{3}} + 3 x^{2} \frac{d ^{3} y}{d x ^{3}} + 45 x^{2} \frac{d ^{2} y}{d x ^{2}} - 2 x \frac{d ^{3} y}{d x ^{3}} + 18 x \frac{d ^{2} y}{d x ^{2}} + 90 x \frac{d y}{d x} + 9 \frac{d ^{3} y}{d x ^{3}} - 6 \frac{d ^{2} y}{d x ^{2}} + 18 \frac{d y}{d x} + 30 y

解答ステップ

\frac{d ^{3}}{d x ^{3}} ((y) (5 x^{3} + 3 x^{2} - 2 x + 9))

\frac{d}{d x} (y (5 x^{3} + 3 x^{2} - 2 x + 9)) = \frac{d y}{d x} (5 x^{3} + 3 x^{2} - 2 x + 9) + (15 x^{2} + 6 x - 2) y

= \frac{d ^{2}}{d x ^{2}} (\frac{d y}{d x} (5 x^{3} + 3 x^{2} - 2 x + 9) + (15 x^{2} + 6 x - 2) y)

\frac{d}{d x} (\frac{d y}{d x} (5 x^{3} + 3 x^{2} - 2 x + 9) + (15 x^{2} + 6 x - 2) y) = \frac{d ^{2} y}{d x ^{2}} (5 x^{3} + 3 x^{2} - 2 x + 9) + 2 \frac{d y}{d x} (15 x^{2} + 6 x - 2) + (30 x + 6) y

= \frac{d}{d x} (\frac{d ^{2} y}{d x ^{2}} (5 x^{3} + 3 x^{2} - 2 x + 9) + 2 \frac{d y}{d x} (15 x^{2} + 6 x - 2) + (30 x + 6) y)

\frac{d}{d x} (\frac{d ^{2} y}{d x ^{2}} (5 x^{3} + 3 x^{2} - 2 x + 9) + 2 \frac{d y}{d x} (15 x^{2} + 6 x - 2) + (30 x + 6) y) = 5 x^{3} \frac{d ^{3} y}{d x ^{3}} + 3 x^{2} \frac{d ^{3} y}{d x ^{3}} + 45 x^{2} \frac{d ^{2} y}{d x ^{2}} - 2 x \frac{d ^{3} y}{d x ^{3}} + 18 x \frac{d ^{2} y}{d x ^{2}} + 90 x \frac{d y}{d x} + 9 \frac{d ^{3} y}{d x ^{3}} - 6 \frac{d ^{2} y}{d x ^{2}} + 18 \frac{d y}{d x} + 30 y

= 5 x^{3} \frac{d ^{3} y}{d x ^{3}} + 3 x^{2} \frac{d ^{3} y}{d x ^{3}} + 45 x^{2} \frac{d ^{2} y}{d x ^{2}} - 2 x \frac{d ^{3} y}{d x ^{3}} + 18 x \frac{d ^{2} y}{d x ^{2}} + 90 x \frac{d y}{d x} + 9 \frac{d ^{3} y}{d x ^{3}} - 6 \frac{d ^{2} y}{d x ^{2}} + 18 \frac{d y}{d x} + 30 y