(d^3)/(dx^3)((y)(2x^4+3x^2+12))

Lösung

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

2 x^{4} \frac{d ^{3} y}{d x ^{3}} + 24 x^{3} \frac{d ^{2} y}{d x ^{2}} + 3 x^{2} \frac{d ^{3} y}{d x ^{3}} + 72 x^{2} \frac{d y}{d x} + 18 x \frac{d ^{2} y}{d x ^{2}} + 48 x y + 12 \frac{d ^{3} y}{d x ^{3}} + 18 \frac{d y}{d x}

Schritte zur Lösung

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

\frac{d}{d x} (y (2 x^{4} + 3 x^{2} + 12)) = \frac{d y}{d x} (2 x^{4} + 3 x^{2} + 12) + (8 x^{3} + 6 x) y

= \frac{d ^{2}}{d x ^{2}} (\frac{d y}{d x} (2 x^{4} + 3 x^{2} + 12) + (8 x^{3} + 6 x) y)

\frac{d}{d x} (\frac{d y}{d x} (2 x^{4} + 3 x^{2} + 12) + (8 x^{3} + 6 x) y) = \frac{d ^{2} y}{d x ^{2}} (2 x^{4} + 3 x^{2} + 12) + 2 (8 x^{3} + 6 x) \frac{d y}{d x} + (24 x^{2} + 6) y

= \frac{d}{d x} (\frac{d ^{2} y}{d x ^{2}} (2 x^{4} + 3 x^{2} + 12) + 2 (8 x^{3} + 6 x) \frac{d y}{d x} + (24 x^{2} + 6) y)

\frac{d}{d x} (\frac{d ^{2} y}{d x ^{2}} (2 x^{4} + 3 x^{2} + 12) + 2 (8 x^{3} + 6 x) \frac{d y}{d x} + (24 x^{2} + 6) y) = 2 x^{4} \frac{d ^{3} y}{d x ^{3}} + 24 x^{3} \frac{d ^{2} y}{d x ^{2}} + 3 x^{2} \frac{d ^{3} y}{d x ^{3}} + 72 x^{2} \frac{d y}{d x} + 18 x \frac{d ^{2} y}{d x ^{2}} + 48 x y + 12 \frac{d ^{3} y}{d x ^{3}} + 18 \frac{d y}{d x}

= 2 x^{4} \frac{d ^{3} y}{d x ^{3}} + 24 x^{3} \frac{d ^{2} y}{d x ^{2}} + 3 x^{2} \frac{d ^{3} y}{d x ^{3}} + 72 x^{2} \frac{d y}{d x} + 18 x \frac{d ^{2} y}{d x ^{2}} + 48 x y + 12 \frac{d ^{3} y}{d x ^{3}} + 18 \frac{d y}{d x}

e x p an d \frac{s + 1}{s ( s - 1 ) ( s ^{2} + 4 s + 16 )}

e x p an d \frac{( 2 a - b ) - 3}{4 - 2 ( 2 a - b ) ^{2}}

e x p an d d x + (\frac{x}{y} - sin (y)) d y

e x p an d (4312 + 2) (332 - 4)

s im pl i f y a + b (a - b)