derivative of (7arcsin(4x)^x)

解

\frac{d}{d x} ((7 arcsin (4 x))^{x})

(ln (7 arcsin (4 x)) + \frac{4 x}{arcsin ( 4 x ) - 16 x ^{2} + 1}) (7 arcsin (4 x))^{x}

解答ステップ

\frac{d}{d x} ((7 arcsin (4 x))^{x})

指数の規則を適用する:

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

(7 arcsin (4 x))^{x} = e^{x l n (7 a r c s i n (4 x))}

= \frac{d}{d x} (e^{x l n (7 a r c s i n (4 x))})

連鎖法則を適用:

e^{x l n (7 a r c s i n (4 x))} \frac{d}{d x} (x ln (7 arcsin (4 x)))

= e^{x l n (7 a r c s i n (4 x))} \frac{d}{d x} (x ln (7 arcsin (4 x)))

\frac{d}{d x} (x ln (7 arcsin (4 x))) = ln (7 arcsin (4 x)) + \frac{4 x}{arcsin ( 4 x ) - 16 x ^{2} + 1}

= e^{x l n (7 a r c s i n (4 x))} (ln (7 arcsin (4 x)) + \frac{4 x}{arcsin ( 4 x ) - 16 x ^{2} + 1})

e^{x l n (7 a r c s i n (4 x))} = (7 arcsin (4 x))^{x}

= (ln (7 arcsin (4 x)) + \frac{4 x}{arcsin ( 4 x ) - 16 x ^{2} + 1}) (7 arcsin (4 x))^{x}