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

Lösung

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

- x^{4} \frac{d ^{4} y}{d x ^{4}} - 16 x^{3} \frac{d ^{3} y}{d x ^{3}} - 72 x^{2} \frac{d ^{2} y}{d x ^{2}} - 96 x \frac{d y}{d x} + 5 x \frac{d ^{4} y}{d x ^{4}} + 5 \frac{d ^{3} y}{d x ^{3}} - 24 y + 15 \frac{d}{d x} (\frac{d ^{2} y}{d x ^{2}})

Schritte zur Lösung

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

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

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

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

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

\frac{d}{d x} (\frac{d ^{2} y}{d x ^{2}} (- x^{4} + 5 x) + 2 (- 4 x^{3} + 5) \frac{d y}{d x} - 12 x^{2} y) = - x^{4} \frac{d ^{3} y}{d x ^{3}} - 12 x^{3} \frac{d ^{2} y}{d x ^{2}} - 36 x^{2} \frac{d y}{d x} + 5 x \frac{d ^{3} y}{d x ^{3}} - 24 x y + 15 \frac{d ^{2} y}{d x ^{2}}

= \frac{d}{d x} (- x^{4} \frac{d ^{3} y}{d x ^{3}} - 12 x^{3} \frac{d ^{2} y}{d x ^{2}} - 36 x^{2} \frac{d y}{d x} + 5 x \frac{d ^{3} y}{d x ^{3}} - 24 x y + 15 \frac{d ^{2} y}{d x ^{2}})

\frac{d}{d x} (- x^{4} \frac{d ^{3} y}{d x ^{3}} - 12 x^{3} \frac{d ^{2} y}{d x ^{2}} - 36 x^{2} \frac{d y}{d x} + 5 x \frac{d ^{3} y}{d x ^{3}} - 24 x y + 15 \frac{d ^{2} y}{d x ^{2}}) = - x^{4} \frac{d ^{4} y}{d x ^{4}} - 16 x^{3} \frac{d ^{3} y}{d x ^{3}} - 72 x^{2} \frac{d ^{2} y}{d x ^{2}} - 96 x \frac{d y}{d x} + 5 x \frac{d ^{4} y}{d x ^{4}} + 5 \frac{d ^{3} y}{d x ^{3}} - 24 y + 15 \frac{d}{d x} (\frac{d ^{2} y}{d x ^{2}})

= - x^{4} \frac{d ^{4} y}{d x ^{4}} - 16 x^{3} \frac{d ^{3} y}{d x ^{3}} - 72 x^{2} \frac{d ^{2} y}{d x ^{2}} - 96 x \frac{d y}{d x} + 5 x \frac{d ^{4} y}{d x ^{4}} + 5 \frac{d ^{3} y}{d x ^{3}} - 24 y + 15 \frac{d}{d x} (\frac{d ^{2} y}{d x ^{2}})

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

e x p an d \frac{( 4 x - 15 ) d x}{( x ^{2} - 4 x + 13 )}

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

e x p an d 2 t (45 cos (3 x) + 16 e^{- 4 x} - 8 sin (2 x))

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