(d^4)/(dt^4)((y)(t^5))

解答

\frac{d ^{4}}{d t ^{4}} ((y) (t^{5}))

t^{5} \frac{d ^{4} y}{d t ^{4}} + 20 t^{4} \frac{d ^{3} y}{d t ^{3}} + 120 t^{3} \frac{d ^{2} y}{d t ^{2}} + 240 t^{2} \frac{d y}{d t} + 120 t y

求解步骤

\frac{d ^{4}}{d t ^{4}} ((y) (t^{5}))

\frac{d}{d t} (y t^{5}) = \frac{d y}{d t} t^{5} + 5 t^{4} y

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

\frac{d}{d t} (\frac{d y}{d t} t^{5} + 5 t^{4} y) = t^{5} \frac{d ^{2} y}{d t ^{2}} + 10 t^{4} \frac{d y}{d t} + 20 t^{3} y

= \frac{d ^{2}}{d t ^{2}} (t^{5} \frac{d ^{2} y}{d t ^{2}} + 10 t^{4} \frac{d y}{d t} + 20 t^{3} y)

\frac{d}{d t} (t^{5} \frac{d ^{2} y}{d t ^{2}} + 10 t^{4} \frac{d y}{d t} + 20 t^{3} y) = t^{5} \frac{d ^{3} y}{d t ^{3}} + 15 t^{4} \frac{d ^{2} y}{d t ^{2}} + 60 t^{3} \frac{d y}{d t} + 60 t^{2} y

= \frac{d}{d t} (t^{5} \frac{d ^{3} y}{d t ^{3}} + 15 t^{4} \frac{d ^{2} y}{d t ^{2}} + 60 t^{3} \frac{d y}{d t} + 60 t^{2} y)

\frac{d}{d t} (t^{5} \frac{d ^{3} y}{d t ^{3}} + 15 t^{4} \frac{d ^{2} y}{d t ^{2}} + 60 t^{3} \frac{d y}{d t} + 60 t^{2} y) = t^{5} \frac{d ^{4} y}{d t ^{4}} + 20 t^{4} \frac{d ^{3} y}{d t ^{3}} + 120 t^{3} \frac{d ^{2} y}{d t ^{2}} + 240 t^{2} \frac{d y}{d t} + 120 t y

= t^{5} \frac{d ^{4} y}{d t ^{4}} + 20 t^{4} \frac{d ^{3} y}{d t ^{3}} + 120 t^{3} \frac{d ^{2} y}{d t ^{2}} + 240 t^{2} \frac{d y}{d t} + 120 t y

f a c t or (5 x + 1)^{3} - 1^{3}

f a c t or (-) n^{3}

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f a c t or 15 x^{2} \cdot x^{2}

f a c t or \frac{\frac{- 13}{7}}{2}