derivative of x^{arcsin(x})

解

\frac{d}{d x} (x^{a r c s i n (x)})

x^{a r c s i n (x)} (\frac{ln ( x )}{1 - x ^{2}} + \frac{arcsin ( x )}{x})

解答ステップ

\frac{d}{d x} (x^{a r c s i n (x)})

指数の規則を適用する:

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

x^{a r c s i n (x)} = e^{a r c s i n (x) l n (x)}

= \frac{d}{d x} (e^{a r c s i n (x) l n (x)})

連鎖法則を適用:

e^{a r c s i n (x) l n (x)} \frac{d}{d x} (arcsin (x) ln (x))

= e^{a r c s i n (x) l n (x)} \frac{d}{d x} (arcsin (x) ln (x))

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

= e^{a r c s i n (x) l n (x)} (\frac{ln ( x )}{1 - x ^{2}} + \frac{arcsin ( x )}{x})

e^{a r c s i n (x) l n (x)} = x^{a r c s i n (x)}

= x^{a r c s i n (x)} (\frac{ln ( x )}{1 - x ^{2}} + \frac{arcsin ( x )}{x})