derivative of x^{1/(x^2})

解

\frac{d}{d x} (x^{\frac{1}{x ^{2}}})

x^{\frac{1 - 3 x ^{2}}{x ^{2}}} (- 2 ln (x) + 1)

解答ステップ

\frac{d}{d x} (x^{\frac{1}{x ^{2}}})

指数の規則を適用する:

a^{b} = e^{b l n (a)}

x^{\frac{1}{x ^{2}}} = e^{\frac{1}{x ^{2}} l n (x)}

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

連鎖法則を適用:

e^{\frac{1}{x ^{2}} l n (x)} \frac{d}{d x} (\frac{1}{x ^{2}} ln (x))

= e^{\frac{1}{x ^{2}} l n (x)} \frac{d}{d x} (\frac{1}{x ^{2}} ln (x))

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

= e^{\frac{1}{x ^{2}} l n (x)} \frac{- 2 ln ( x ) + 1}{x ^{3}}

簡素化

e^{\frac{1}{x ^{2}} l n (x)} \frac{- 2 ln ( x ) + 1}{x ^{3}} : x^{\frac{1 - 3 x ^{2}}{x ^{2}}} (- 2 ln (x) + 1)

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