derivative sqrt(1-x^2)arcsin(x)

解

導関数

1 - x^{2} arcsin (x)

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

解答ステップ

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

積の法則を適用:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = 1 - x^{2}, g = arcsin (x)

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

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

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

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

簡素化

(- \frac{x}{1 - x ^{2}}) arcsin (x) + \frac{1}{1 - x ^{2}} 1 - x^{2} : - \frac{x arcsin ( x )}{1 - x ^{2}} + 1

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