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

解答

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

x^{3} \frac{d ^{3} y}{d x ^{3}} + 3 x^{2} \frac{d ^{3} y}{d x ^{3}} + 9 x^{2} \frac{d ^{2} y}{d x ^{2}} + 6 x \frac{d ^{3} y}{d x ^{3}} + 18 x \frac{d ^{2} y}{d x ^{2}} + 18 x \frac{d y}{d x} + 18 \frac{d ^{2} y}{d x ^{2}} + 18 \frac{d y}{d x} + 6 y

求解步骤

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

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

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

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

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

\frac{d}{d x} (\frac{d ^{2} y}{d x ^{2}} (x^{3} + 3 x^{2} + 6 x) + 2 \frac{d y}{d x} (3 x^{2} + 6 x + 6) + (6 x + 6) y) = x^{3} \frac{d ^{3} y}{d x ^{3}} + 3 x^{2} \frac{d ^{3} y}{d x ^{3}} + 9 x^{2} \frac{d ^{2} y}{d x ^{2}} + 6 x \frac{d ^{3} y}{d x ^{3}} + 18 x \frac{d ^{2} y}{d x ^{2}} + 18 x \frac{d y}{d x} + 18 \frac{d ^{2} y}{d x ^{2}} + 18 \frac{d y}{d x} + 6 y

= x^{3} \frac{d ^{3} y}{d x ^{3}} + 3 x^{2} \frac{d ^{3} y}{d x ^{3}} + 9 x^{2} \frac{d ^{2} y}{d x ^{2}} + 6 x \frac{d ^{3} y}{d x ^{3}} + 18 x \frac{d ^{2} y}{d x ^{2}} + 18 x \frac{d y}{d x} + 18 \frac{d ^{2} y}{d x ^{2}} + 18 \frac{d y}{d x} + 6 y

e x p an d (x \cdot y = 5, x + y = 7) x + y

e x p an d \frac{( n + 1 ) ( n + 2 )}{2}

s im pl i f y (sec (x) + csc (x)) (sin (x) + cos (x)) - 2

s im pl i f y (1 - x - 2) (1 - x + 2)

e x p an d (x - y) l e f t (x + y)