derivative y=x^{1/x}

解答

导数

y = x^{\frac{1}{x}}

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

求解步骤

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

使用指数法则:

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

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

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

使用连锁律:

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

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

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

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

化简

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

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

\frac{\partial}{\partial x} (\frac{2 x}{x ^{2} + y ^{2} + z ^{2}})

x \to + 0 lim (\frac{( 1 + x ) ^{4} - 1}{x})

\frac{\partial}{\partial y} (x y + yz + z x)

x \to 5 lim (\frac{f ( x )}{g ( x )})

\int_{0}^{2} 6 - x^{2} d x