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

解

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

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

解答ステップ

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

指数の規則を適用する:

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

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

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

連鎖法則を適用:

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

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

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

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

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

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