(d^3)/(dx^3)((y)(4x^5-3x^4+2x))

Lösung

\frac{d ^{3}}{d x ^{3}} ((y) (4 x^{5} - 3 x^{4} + 2 x))

4 x^{5} \frac{d ^{3} y}{d x ^{3}} - 3 x^{4} \frac{d ^{3} y}{d x ^{3}} + 60 x^{4} \frac{d ^{2} y}{d x ^{2}} - 36 x^{3} \frac{d ^{2} y}{d x ^{2}} + 240 x^{3} \frac{d y}{d x} - 108 x^{2} \frac{d y}{d x} + 240 x^{2} y + 2 x \frac{d ^{3} y}{d x ^{3}} - 72 x y + 6 \frac{d ^{2} y}{d x ^{2}}

Schritte zur Lösung

\frac{d ^{3}}{d x ^{3}} ((y) (4 x^{5} - 3 x^{4} + 2 x))

\frac{d}{d x} (y (4 x^{5} - 3 x^{4} + 2 x)) = \frac{d y}{d x} (4 x^{5} - 3 x^{4} + 2 x) + (20 x^{4} - 12 x^{3} + 2) y

= \frac{d ^{2}}{d x ^{2}} (\frac{d y}{d x} (4 x^{5} - 3 x^{4} + 2 x) + (20 x^{4} - 12 x^{3} + 2) y)

\frac{d}{d x} (\frac{d y}{d x} (4 x^{5} - 3 x^{4} + 2 x) + (20 x^{4} - 12 x^{3} + 2) y) = \frac{d ^{2} y}{d x ^{2}} (4 x^{5} - 3 x^{4} + 2 x) + 2 \frac{d y}{d x} (20 x^{4} - 12 x^{3} + 2) + (80 x^{3} - 36 x^{2}) y

= \frac{d}{d x} (\frac{d ^{2} y}{d x ^{2}} (4 x^{5} - 3 x^{4} + 2 x) + 2 \frac{d y}{d x} (20 x^{4} - 12 x^{3} + 2) + (80 x^{3} - 36 x^{2}) y)

\frac{d}{d x} (\frac{d ^{2} y}{d x ^{2}} (4 x^{5} - 3 x^{4} + 2 x) + 2 \frac{d y}{d x} (20 x^{4} - 12 x^{3} + 2) + (80 x^{3} - 36 x^{2}) y) = 4 x^{5} \frac{d ^{3} y}{d x ^{3}} - 3 x^{4} \frac{d ^{3} y}{d x ^{3}} + 60 x^{4} \frac{d ^{2} y}{d x ^{2}} - 36 x^{3} \frac{d ^{2} y}{d x ^{2}} + 240 x^{3} \frac{d y}{d x} - 108 x^{2} \frac{d y}{d x} + 240 x^{2} y + 2 x \frac{d ^{3} y}{d x ^{3}} - 72 x y + 6 \frac{d ^{2} y}{d x ^{2}}

= 4 x^{5} \frac{d ^{3} y}{d x ^{3}} - 3 x^{4} \frac{d ^{3} y}{d x ^{3}} + 60 x^{4} \frac{d ^{2} y}{d x ^{2}} - 36 x^{3} \frac{d ^{2} y}{d x ^{2}} + 240 x^{3} \frac{d y}{d x} - 108 x^{2} \frac{d y}{d x} + 240 x^{2} y + 2 x \frac{d ^{3} y}{d x ^{3}} - 72 x y + 6 \frac{d ^{2} y}{d x ^{2}}

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

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

e x p an d d x (2 x + 2)

e x p an d (- 2 e^{x}) (1 + e^{x})^{3}

e x p an d \frac{( 6 - z ) ^{3}}{3}