derivative x^{x/3}

解答

导数

x^{\frac{x}{3}}

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

求解步骤

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

使用指数法则:

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

x^{\frac{x}{3}} = e^{\frac{x}{3} l n (x)}

= \frac{d}{d x} (e^{\frac{x}{3} l n (x)})

使用连锁律:

e^{\frac{x}{3} l n (x)} \frac{d}{d x} (\frac{x}{3} ln (x))

= e^{\frac{x}{3} l n (x)} \frac{d}{d x} (\frac{x}{3} ln (x))

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

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

化简

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

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

f^{^{'}} (x) = 6 x^{2}

n = 1 \sum \infty \frac{2 ^{n}}{4 ^{n}}

x \to 0 lim (\frac{1 - cos ( 3 x )}{1 - cos ( 7 x )})

x \to 3 + lim (x^{2} - 9)

(x, y) \to (0, 2) lim (x^{3} y)