(d^4)/(dx^4)(e^{1/x})

解答

\frac{d ^{4}}{d x ^{4}} (e^{\frac{1}{x}})

- \frac{- 24 e ^{\frac{1}{x}} x ^{3} - 36 e ^{\frac{1}{x}} x ^{2} - 12 e ^{\frac{1}{x}} x - e ^{\frac{1}{x}}}{x ^{8}}

求解步骤

\frac{d ^{4}}{d x ^{4}} (e^{\frac{1}{x}})

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

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

\frac{d}{d x} (- \frac{e ^{\frac{1}{x}}}{x ^{2}}) = - \frac{- e ^{\frac{1}{x}} - 2 e ^{\frac{1}{x}} x}{x ^{4}}

= \frac{d ^{2}}{d x ^{2}} (- \frac{- e ^{\frac{1}{x}} - 2 e ^{\frac{1}{x}} x}{x ^{4}})

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

= \frac{d}{d x} (- \frac{6 e ^{\frac{1}{x}} x ^{2} + e ^{\frac{1}{x}} + 6 e ^{\frac{1}{x}} x}{x ^{6}})

\frac{d}{d x} (- \frac{6 e ^{\frac{1}{x}} x ^{2} + e ^{\frac{1}{x}} + 6 e ^{\frac{1}{x}} x}{x ^{6}}) = - \frac{- 24 e ^{\frac{1}{x}} x ^{3} - 36 e ^{\frac{1}{x}} x ^{2} - 12 e ^{\frac{1}{x}} x - e ^{\frac{1}{x}}}{x ^{8}}

= - \frac{- 24 e ^{\frac{1}{x}} x ^{3} - 36 e ^{\frac{1}{x}} x ^{2} - 12 e ^{\frac{1}{x}} x - e ^{\frac{1}{x}}}{x ^{8}}