(d^3)/(dx^3)((y)(x+1)^{-1})

解

\frac{d ^{3}}{d x ^{3}} ((y) (x + 1)^{- 1})

\frac{x ^{3} \frac{d ^{3} y}{d x ^{3}} + x ^{2} \frac{d ^{3} y}{d x ^{3}} - x ^{2} \frac{d ^{2} y}{d x ^{2}} - 2 x ^{2} \frac{d ^{4} y}{d x ^{4}} + 2 x ^{2} \frac{d ^{5} y}{d x ^{5}} + 2 x \frac{d ^{2}}{d x ^{2}} ( \frac{d y}{d x} ) + 2 x \frac{d ^{4} y}{d x ^{4}} - 5 x \frac{d ^{3} y}{d x ^{3}} - 4 x \frac{d ^{2} y}{d x ^{2}} + 2 x \frac{d ^{5} y}{d x ^{5}} + 6 x \frac{d y}{d x} + 2 \frac{d ^{2}}{d x ^{2}} ( \frac{d y}{d x} ) + 4 \frac{d ^{4} y}{d x ^{4}} - 5 \frac{d ^{3} y}{d x ^{3}} - 3 \frac{d ^{2} y}{d x ^{2}} + 6 \frac{d y}{d x} - 6 y}{( x + 1 ) ^{4}}

解答ステップ

\frac{d ^{3}}{d x ^{3}} ((y) (x + 1)^{- 1})

\frac{d}{d x} (y (x + 1)^{- 1}) = \frac{\frac{d y}{d x}}{x + 1} - \frac{y}{( x + 1 ) ^{2}}

= \frac{d ^{2}}{d x ^{2}} (\frac{\frac{d y}{d x}}{x + 1} - \frac{y}{( x + 1 ) ^{2}})

\frac{d}{d x} (\frac{\frac{d y}{d x}}{x + 1} - \frac{y}{( x + 1 ) ^{2}}) = \frac{x ^{2} \frac{d ^{2} y}{d x ^{2}} + 2 x \frac{d ^{2} y}{d x ^{2}} - 2 x \frac{d y}{d x} + \frac{d ^{2} y}{d x ^{2}} - 2 \frac{d y}{d x} + 2 y}{( x + 1 ) ^{3}}

= \frac{d}{d x} (\frac{x ^{2} \frac{d ^{2} y}{d x ^{2}} + 2 x \frac{d ^{2} y}{d x ^{2}} - 2 x \frac{d y}{d x} + \frac{d ^{2} y}{d x ^{2}} - 2 \frac{d y}{d x} + 2 y}{( x + 1 ) ^{3}})

\frac{d}{d x} (\frac{x ^{2} \frac{d ^{2} y}{d x ^{2}} + 2 x \frac{d ^{2} y}{d x ^{2}} - 2 x \frac{d y}{d x} + \frac{d ^{2} y}{d x ^{2}} - 2 \frac{d y}{d x} + 2 y}{( x + 1 ) ^{3}}) = \frac{x ^{3} \frac{d ^{3} y}{d x ^{3}} + x ^{2} \frac{d ^{3} y}{d x ^{3}} - x ^{2} \frac{d ^{2} y}{d x ^{2}} - 2 x ^{2} \frac{d ^{4} y}{d x ^{4}} + 2 x ^{2} \frac{d ^{5} y}{d x ^{5}} + 2 x \frac{d ^{2}}{d x ^{2}} ( \frac{d y}{d x} ) + 2 x \frac{d ^{4} y}{d x ^{4}} - 5 x \frac{d ^{3} y}{d x ^{3}} - 4 x \frac{d ^{2} y}{d x ^{2}} + 2 x \frac{d ^{5} y}{d x ^{5}} + 6 x \frac{d y}{d x} + 2 \frac{d ^{2}}{d x ^{2}} ( \frac{d y}{d x} ) + 4 \frac{d ^{4} y}{d x ^{4}} - 5 \frac{d ^{3} y}{d x ^{3}} - 3 \frac{d ^{2} y}{d x ^{2}} + 6 \frac{d y}{d x} - 6 y}{( x + 1 ) ^{4}}

= \frac{x ^{3} \frac{d ^{3} y}{d x ^{3}} + x ^{2} \frac{d ^{3} y}{d x ^{3}} - x ^{2} \frac{d ^{2} y}{d x ^{2}} - 2 x ^{2} \frac{d ^{4} y}{d x ^{4}} + 2 x ^{2} \frac{d ^{5} y}{d x ^{5}} + 2 x \frac{d ^{2}}{d x ^{2}} ( \frac{d y}{d x} ) + 2 x \frac{d ^{4} y}{d x ^{4}} - 5 x \frac{d ^{3} y}{d x ^{3}} - 4 x \frac{d ^{2} y}{d x ^{2}} + 2 x \frac{d ^{5} y}{d x ^{5}} + 6 x \frac{d y}{d x} + 2 \frac{d ^{2}}{d x ^{2}} ( \frac{d y}{d x} ) + 4 \frac{d ^{4} y}{d x ^{4}} - 5 \frac{d ^{3} y}{d x ^{3}} - 3 \frac{d ^{2} y}{d x ^{2}} + 6 \frac{d y}{d x} - 6 y}{( x + 1 ) ^{4}}