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

解

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

1536 x \frac{d y}{d x} + (2 x + 1)^{4} \frac{d ^{4} y}{d x ^{4}} + 32 (2 x + 1)^{3} \frac{d ^{3} y}{d x ^{3}} + 288 (2 x + 1)^{2} \frac{d ^{2} y}{d x ^{2}} + 768 \frac{d y}{d x} + 384 y

解答ステップ

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

\frac{d}{d x} (y (2 x + 1)^{4}) = \frac{d y}{d x} (2 x + 1)^{4} + 8 (2 x + 1)^{3} y

= \frac{d ^{3}}{d x ^{3}} (\frac{d y}{d x} (2 x + 1)^{4} + 8 (2 x + 1)^{3} y)

\frac{d}{d x} (\frac{d y}{d x} (2 x + 1)^{4} + 8 (2 x + 1)^{3} y) = \frac{d ^{2} y}{d x ^{2}} (2 x + 1)^{4} + 16 \frac{d y}{d x} (2 x + 1)^{3} + 48 (2 x + 1)^{2} y

= \frac{d ^{2}}{d x ^{2}} (\frac{d ^{2} y}{d x ^{2}} (2 x + 1)^{4} + 16 \frac{d y}{d x} (2 x + 1)^{3} + 48 (2 x + 1)^{2} y)

\frac{d}{d x} (\frac{d ^{2} y}{d x ^{2}} (2 x + 1)^{4} + 16 \frac{d y}{d x} (2 x + 1)^{3} + 48 (2 x + 1)^{2} y) = \frac{d ^{3} y}{d x ^{3}} (2 x + 1)^{4} + 24 \frac{d ^{2} y}{d x ^{2}} (2 x + 1)^{3} + 144 \frac{d y}{d x} (2 x + 1)^{2} + 384 x y + 192 y

= \frac{d}{d x} (\frac{d ^{3} y}{d x ^{3}} (2 x + 1)^{4} + 24 \frac{d ^{2} y}{d x ^{2}} (2 x + 1)^{3} + 144 \frac{d y}{d x} (2 x + 1)^{2} + 384 x y + 192 y)

\frac{d}{d x} (\frac{d ^{3} y}{d x ^{3}} (2 x + 1)^{4} + 24 \frac{d ^{2} y}{d x ^{2}} (2 x + 1)^{3} + 144 \frac{d y}{d x} (2 x + 1)^{2} + 384 x y + 192 y) = 1536 x \frac{d y}{d x} + (2 x + 1)^{4} \frac{d ^{4} y}{d x ^{4}} + 32 (2 x + 1)^{3} \frac{d ^{3} y}{d x ^{3}} + 288 (2 x + 1)^{2} \frac{d ^{2} y}{d x ^{2}} + 768 \frac{d y}{d x} + 384 y

= 1536 x \frac{d y}{d x} + (2 x + 1)^{4} \frac{d ^{4} y}{d x ^{4}} + 32 (2 x + 1)^{3} \frac{d ^{3} y}{d x ^{3}} + 288 (2 x + 1)^{2} \frac{d ^{2} y}{d x ^{2}} + 768 \frac{d y}{d x} + 384 y