(d^3)/(dx^3)((y)(6x-4x^3+2-x^4))

解答

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

- x^{4} \frac{d ^{3} y}{d x ^{3}} - 4 x^{3} \frac{d ^{3} y}{d x ^{3}} - 12 x^{3} \frac{d ^{2} y}{d x ^{2}} - 36 x^{2} \frac{d ^{2} y}{d x ^{2}} - 36 x^{2} \frac{d y}{d x} + 6 x \frac{d ^{3} y}{d x ^{3}} - 72 x \frac{d y}{d x} - 24 x y + 2 \frac{d ^{3} y}{d x ^{3}} + 18 \frac{d ^{2} y}{d x ^{2}} - 24 y

求解步骤

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

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

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

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

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

\frac{d}{d x} (\frac{d ^{2} y}{d x ^{2}} (6 x - 4 x^{3} + 2 - x^{4}) + 2 (- 4 x^{3} - 12 x^{2} + 6) \frac{d y}{d x} + (- 12 x^{2} - 24 x) y) = - x^{4} \frac{d ^{3} y}{d x ^{3}} - 4 x^{3} \frac{d ^{3} y}{d x ^{3}} - 12 x^{3} \frac{d ^{2} y}{d x ^{2}} - 36 x^{2} \frac{d ^{2} y}{d x ^{2}} - 36 x^{2} \frac{d y}{d x} + 6 x \frac{d ^{3} y}{d x ^{3}} - 72 x \frac{d y}{d x} - 24 x y + 2 \frac{d ^{3} y}{d x ^{3}} + 18 \frac{d ^{2} y}{d x ^{2}} - 24 y

= - x^{4} \frac{d ^{3} y}{d x ^{3}} - 4 x^{3} \frac{d ^{3} y}{d x ^{3}} - 12 x^{3} \frac{d ^{2} y}{d x ^{2}} - 36 x^{2} \frac{d ^{2} y}{d x ^{2}} - 36 x^{2} \frac{d y}{d x} + 6 x \frac{d ^{3} y}{d x ^{3}} - 72 x \frac{d y}{d x} - 24 x y + 2 \frac{d ^{3} y}{d x ^{3}} + 18 \frac{d ^{2} y}{d x ^{2}} - 24 y

e x p an d \frac{1}{s ( a ^{2} + w ^{2} )}

e x p an d (t + e^{t - 2}) U (t - 2)

e x p an d \frac{x}{( x - 2 ) ( x - 4 )}

s im pl i f y (x - 37)^{2}

e x p an d \frac{[ ( x + 3 ) ( x + 4 ) ( x + 7 ) ( x + 9 ) ]}{( x + 6 )}