(d^4}{dx^4}((y)(3/(x^2)-\frac{x^2)/3))

解答

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

\frac{- x ^{8} \frac{d ^{4} y}{d x ^{4}} - 8 x ^{7} \frac{d ^{3} y}{d x ^{3}} - 12 x ^{6} \frac{d ^{2} y}{d x ^{2}} + 9 x ^{4} \frac{d ^{4} y}{d x ^{4}} - 72 x ^{3} \frac{d ^{3} y}{d x ^{3}} + 324 x ^{2} \frac{d ^{2} y}{d x ^{2}} - 864 x \frac{d y}{d x} + 1080 y}{3 x ^{6}}

求解步骤

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

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

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

\frac{d}{d x} (\frac{3 \frac{d y}{d x}}{x ^{2}} + \frac{- x ^{2} \frac{d y}{d x} - 2 x y}{3} - \frac{6 y}{x ^{3}}) = \frac{- x ^{6} \frac{d ^{2} y}{d x ^{2}} - 4 x ^{5} \frac{d y}{d x} - 2 x ^{4} y + 9 x ^{2} \frac{d ^{2} y}{d x ^{2}} - 36 x \frac{d y}{d x} + 54 y}{3 x ^{4}}

= \frac{d ^{2}}{d x ^{2}} (\frac{- x ^{6} \frac{d ^{2} y}{d x ^{2}} - 4 x ^{5} \frac{d y}{d x} - 2 x ^{4} y + 9 x ^{2} \frac{d ^{2} y}{d x ^{2}} - 36 x \frac{d y}{d x} + 54 y}{3 x ^{4}})

\frac{d}{d x} (\frac{- x ^{6} \frac{d ^{2} y}{d x ^{2}} - 4 x ^{5} \frac{d y}{d x} - 2 x ^{4} y + 9 x ^{2} \frac{d ^{2} y}{d x ^{2}} - 36 x \frac{d y}{d x} + 54 y}{3 x ^{4}}) = \frac{- x ^{7} \frac{d ^{3} y}{d x ^{3}} - 6 x ^{6} \frac{d ^{2} y}{d x ^{2}} - 6 x ^{5} \frac{d y}{d x} + 9 x ^{3} \frac{d ^{3} y}{d x ^{3}} - 54 x ^{2} \frac{d ^{2} y}{d x ^{2}} + 162 x \frac{d y}{d x} - 216 y}{3 x ^{5}}

= \frac{d}{d x} (\frac{- x ^{7} \frac{d ^{3} y}{d x ^{3}} - 6 x ^{6} \frac{d ^{2} y}{d x ^{2}} - 6 x ^{5} \frac{d y}{d x} + 9 x ^{3} \frac{d ^{3} y}{d x ^{3}} - 54 x ^{2} \frac{d ^{2} y}{d x ^{2}} + 162 x \frac{d y}{d x} - 216 y}{3 x ^{5}})

\frac{d}{d x} (\frac{- x ^{7} \frac{d ^{3} y}{d x ^{3}} - 6 x ^{6} \frac{d ^{2} y}{d x ^{2}} - 6 x ^{5} \frac{d y}{d x} + 9 x ^{3} \frac{d ^{3} y}{d x ^{3}} - 54 x ^{2} \frac{d ^{2} y}{d x ^{2}} + 162 x \frac{d y}{d x} - 216 y}{3 x ^{5}}) = \frac{- x ^{8} \frac{d ^{4} y}{d x ^{4}} - 8 x ^{7} \frac{d ^{3} y}{d x ^{3}} - 12 x ^{6} \frac{d ^{2} y}{d x ^{2}} + 9 x ^{4} \frac{d ^{4} y}{d x ^{4}} - 72 x ^{3} \frac{d ^{3} y}{d x ^{3}} + 324 x ^{2} \frac{d ^{2} y}{d x ^{2}} - 864 x \frac{d y}{d x} + 1080 y}{3 x ^{6}}

= \frac{- x ^{8} \frac{d ^{4} y}{d x ^{4}} - 8 x ^{7} \frac{d ^{3} y}{d x ^{3}} - 12 x ^{6} \frac{d ^{2} y}{d x ^{2}} + 9 x ^{4} \frac{d ^{4} y}{d x ^{4}} - 72 x ^{3} \frac{d ^{3} y}{d x ^{3}} + 324 x ^{2} \frac{d ^{2} y}{d x ^{2}} - 864 x \frac{d y}{d x} + 1080 y}{3 x ^{6}}

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