(d^4)/(dx^4)((y)(2ln(x)))

解

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

\frac{2 ( x ^{4} ln ( x ) \frac{d ^{4} y}{d x ^{4}} + 4 x ^{3} \frac{d ^{3} y}{d x ^{3}} - 6 x ^{2} \frac{d ^{2} y}{d x ^{2}} + 8 x \frac{d y}{d x} - 6 y )}{x ^{4}}

解答ステップ

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

\frac{d}{d x} (y \cdot 2 ln (x)) = 2 (ln (x) \frac{d y}{d x} + \frac{y}{x})

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

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

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

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

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

\frac{d}{d x} \frac{2 ( x ^{3} ln ( x ) \frac{d ^{3} y}{d x ^{3}} + 3 x ^{2} \frac{d ^{2} y}{d x ^{2}} - 3 x \frac{d y}{d x} + 2 y )}{x ^{3}} = \frac{2 ( x ^{4} ln ( x ) \frac{d ^{4} y}{d x ^{4}} + 4 x ^{3} \frac{d ^{3} y}{d x ^{3}} - 6 x ^{2} \frac{d ^{2} y}{d x ^{2}} + 8 x \frac{d y}{d x} - 6 y )}{x ^{4}}

= \frac{2 ( x ^{4} ln ( x ) \frac{d ^{4} y}{d x ^{4}} + 4 x ^{3} \frac{d ^{3} y}{d x ^{3}} - 6 x ^{2} \frac{d ^{2} y}{d x ^{2}} + 8 x \frac{d y}{d x} - 6 y )}{x ^{4}}