(d^3)/(dx^3)((y)(5x^4-x^3+2x))

解

\frac{d ^{3}}{d x ^{3}} ((y) (5 x^{4} - x^{3} + 2 x))

5 x^{4} \frac{d ^{3} y}{d x ^{3}} - x^{3} \frac{d ^{3} y}{d x ^{3}} + 60 x^{3} \frac{d ^{2} y}{d x ^{2}} - 9 x^{2} \frac{d ^{2} y}{d x ^{2}} + 180 x^{2} \frac{d y}{d x} + 2 x \frac{d ^{3} y}{d x ^{3}} - 18 x \frac{d y}{d x} + 120 x y + 6 \frac{d ^{2} y}{d x ^{2}} - 6 y

解答ステップ

\frac{d ^{3}}{d x ^{3}} ((y) (5 x^{4} - x^{3} + 2 x))

\frac{d}{d x} (y (5 x^{4} - x^{3} + 2 x)) = \frac{d y}{d x} (5 x^{4} - x^{3} + 2 x) + (20 x^{3} - 3 x^{2} + 2) y

= \frac{d ^{2}}{d x ^{2}} (\frac{d y}{d x} (5 x^{4} - x^{3} + 2 x) + (20 x^{3} - 3 x^{2} + 2) y)

\frac{d}{d x} (\frac{d y}{d x} (5 x^{4} - x^{3} + 2 x) + (20 x^{3} - 3 x^{2} + 2) y) = \frac{d ^{2} y}{d x ^{2}} (5 x^{4} - x^{3} + 2 x) + 2 \frac{d y}{d x} (20 x^{3} - 3 x^{2} + 2) + (60 x^{2} - 6 x) y

= \frac{d}{d x} (\frac{d ^{2} y}{d x ^{2}} (5 x^{4} - x^{3} + 2 x) + 2 \frac{d y}{d x} (20 x^{3} - 3 x^{2} + 2) + (60 x^{2} - 6 x) y)

\frac{d}{d x} (\frac{d ^{2} y}{d x ^{2}} (5 x^{4} - x^{3} + 2 x) + 2 \frac{d y}{d x} (20 x^{3} - 3 x^{2} + 2) + (60 x^{2} - 6 x) y) = 5 x^{4} \frac{d ^{3} y}{d x ^{3}} - x^{3} \frac{d ^{3} y}{d x ^{3}} + 60 x^{3} \frac{d ^{2} y}{d x ^{2}} - 9 x^{2} \frac{d ^{2} y}{d x ^{2}} + 180 x^{2} \frac{d y}{d x} + 2 x \frac{d ^{3} y}{d x ^{3}} - 18 x \frac{d y}{d x} + 120 x y + 6 \frac{d ^{2} y}{d x ^{2}} - 6 y

= 5 x^{4} \frac{d ^{3} y}{d x ^{3}} - x^{3} \frac{d ^{3} y}{d x ^{3}} + 60 x^{3} \frac{d ^{2} y}{d x ^{2}} - 9 x^{2} \frac{d ^{2} y}{d x ^{2}} + 180 x^{2} \frac{d y}{d x} + 2 x \frac{d ^{3} y}{d x ^{3}} - 18 x \frac{d y}{d x} + 120 x y + 6 \frac{d ^{2} y}{d x ^{2}} - 6 y