(d^3)/(dx^3)((y)(x^2-1)(2x+3))

解答

\frac{d ^{3}}{d x ^{3}} ((y) (x^{2} - 1) (2 x + 3))

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

求解步骤

\frac{d ^{3}}{d x ^{3}} ((y) (x^{2} - 1) (2 x + 3))

\frac{d}{d x} (y (x^{2} - 1) (2 x + 3)) = \frac{d y}{d x} (x^{2} - 1) (2 x + 3) + (6 x^{2} + 6 x - 2) y

= \frac{d ^{2}}{d x ^{2}} (\frac{d y}{d x} (x^{2} - 1) (2 x + 3) + (6 x^{2} + 6 x - 2) y)

\frac{d}{d x} (\frac{d y}{d x} (x^{2} - 1) (2 x + 3) + (6 x^{2} + 6 x - 2) y) = \frac{d ^{2} y}{d x ^{2}} (x^{2} - 1) (2 x + 3) + 2 \frac{d y}{d x} (6 x^{2} + 6 x - 2) + (12 x + 6) y

= \frac{d}{d x} (\frac{d ^{2} y}{d x ^{2}} (x^{2} - 1) (2 x + 3) + 2 \frac{d y}{d x} (6 x^{2} + 6 x - 2) + (12 x + 6) y)

\frac{d}{d x} (\frac{d ^{2} y}{d x ^{2}} (x^{2} - 1) (2 x + 3) + 2 \frac{d y}{d x} (6 x^{2} + 6 x - 2) + (12 x + 6) y) = 2 x^{3} \frac{d ^{3} y}{d x ^{3}} + 3 x^{2} \frac{d ^{3} y}{d x ^{3}} + 18 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}} + 36 x \frac{d y}{d x} - 3 \frac{d ^{3} y}{d x ^{3}} - 6 \frac{d ^{2} y}{d x ^{2}} + 18 \frac{d y}{d x} + 12 y

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

e x p an d (4 y + 2 x - 5) d x + (6 y + 4 x - 1) d y

e x p an d - \frac{8}{n ^{3}} (\frac{n ( n + 1 )}{2})^{2}

e x p an d (2 x^{2} y + \frac{1}{x y ^{2}})^{7}

e x p an d \frac{1}{( 1 - x ^{4} )} \cdot \frac{1}{( 1 - x ^{5} )}

e x p an d sin^{2} (u) (v^{2} + 1) + cos^{2} (u) (v^{2} + 1) + v^{2}