(d^3)/(dx^3)((y)(sqrt(x)+\sqrt[3]{x}))

解答

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

\frac{72 x ^{3} \frac{d ^{2} y}{d x ^{2}} - 144 x ^{2} \frac{d y}{d x} + 80 x y + 324 x ^{\frac{19}{6}} \frac{d ^{2} y}{d x ^{2}} - 162 x ^{\frac{13}{6}} \frac{d y}{d x} + 216 x ^{\frac{11}{3}} x \frac{d ^{3} y}{d x ^{3}} + 216 x ^{\frac{11}{3}} 3 x \frac{d ^{3} y}{d x ^{3}} + 144 x ^{\frac{8}{3}} 3 x \frac{d ^{2} y}{d x ^{2}} + 81 x ^{\frac{7}{6}} y}{216 x ^{\frac{11}{3}}}

求解步骤

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

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

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

\frac{d}{d x} (\frac{d y}{d x} (x + 3 x) + (\frac{1}{2 x} + \frac{1}{3 x ^{\frac{2}{3}}}) y) = x \frac{d ^{2} y}{d x ^{2}} + 3 x \frac{d ^{2} y}{d x ^{2}} + \frac{\frac{d y}{d x}}{x} + \frac{2 \frac{d y}{d x}}{3 x ^{\frac{2}{3}}} - \frac{y}{4 x ^{\frac{3}{2}}} - \frac{2 y}{9 x ^{\frac{5}{3}}}

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

\frac{d}{d x} (x \frac{d ^{2} y}{d x ^{2}} + 3 x \frac{d ^{2} y}{d x ^{2}} + \frac{\frac{d y}{d x}}{x} + \frac{2 \frac{d y}{d x}}{3 x ^{\frac{2}{3}}} - \frac{y}{4 x ^{\frac{3}{2}}} - \frac{2 y}{9 x ^{\frac{5}{3}}}) = \frac{72 x ^{3} \frac{d ^{2} y}{d x ^{2}} - 144 x ^{2} \frac{d y}{d x} + 80 x y + 324 x ^{\frac{19}{6}} \frac{d ^{2} y}{d x ^{2}} - 162 x ^{\frac{13}{6}} \frac{d y}{d x} + 216 x ^{\frac{11}{3}} x \frac{d ^{3} y}{d x ^{3}} + 216 x ^{\frac{11}{3}} 3 x \frac{d ^{3} y}{d x ^{3}} + 144 x ^{\frac{8}{3}} 3 x \frac{d ^{2} y}{d x ^{2}} + 81 x ^{\frac{7}{6}} y}{216 x ^{\frac{11}{3}}}

= \frac{72 x ^{3} \frac{d ^{2} y}{d x ^{2}} - 144 x ^{2} \frac{d y}{d x} + 80 x y + 324 x ^{\frac{19}{6}} \frac{d ^{2} y}{d x ^{2}} - 162 x ^{\frac{13}{6}} \frac{d y}{d x} + 216 x ^{\frac{11}{3}} x \frac{d ^{3} y}{d x ^{3}} + 216 x ^{\frac{11}{3}} 3 x \frac{d ^{3} y}{d x ^{3}} + 144 x ^{\frac{8}{3}} 3 x \frac{d ^{2} y}{d x ^{2}} + 81 x ^{\frac{7}{6}} y}{216 x ^{\frac{11}{3}}}

e x p an d \frac{( m - 2 k + 5 ) ( m - 2 k + 4 )}{( k - 3 ) ( k - 2 )}

e x p an d \frac{e ^{- s} + 4 s - 28}{s ( - 7 + s )}

s im pl i f y (5 - i)^{4}

e x p an d (x + \frac{5}{4})^{2}

e x p an d 3 l e f t (m + 1) - 2