derivative of (arcsin(5x)^{2x})

解

\frac{d}{d x} ((arcsin (5 x))^{2 x})

2 arcsin^{2 x} (5 x) (ln (arcsin (5 x)) + \frac{5 x}{arcsin ( 5 x ) 1 - 25 x ^{2}})

解答ステップ

\frac{d}{d x} ((arcsin (5 x))^{2 x})

指数の規則を適用する:

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

(arcsin (5 x))^{2 x} = e^{2 x l n (a r c s i n (5 x))}

= \frac{d}{d x} (e^{2 x l n (a r c s i n (5 x))})

連鎖法則を適用:

e^{2 x l n (a r c s i n (5 x))} \frac{d}{d x} (2 x ln (arcsin (5 x)))

= e^{2 x l n (a r c s i n (5 x))} \frac{d}{d x} (2 x ln (arcsin (5 x)))

\frac{d}{d x} (2 x ln (arcsin (5 x))) = 2 (ln (arcsin (5 x)) + \frac{5 x}{arcsin ( 5 x ) 1 - 25 x ^{2}})

= e^{2 x l n (a r c s i n (5 x))} \cdot 2 (ln (arcsin (5 x)) + \frac{5 x}{arcsin ( 5 x ) 1 - 25 x ^{2}})

e^{2 x l n (a r c s i n (5 x))} = arcsin^{2 x} (5 x)

= 2 arcsin^{2 x} (5 x) (ln (arcsin (5 x)) + \frac{5 x}{arcsin ( 5 x ) 1 - 25 x ^{2}})