derivative of arctan(arcsin(sqrt(x)))

解

\frac{d}{d x} (arctan (arcsin (x)))

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

解答ステップ

\frac{d}{d x} (arctan (arcsin (x)))

連鎖法則を適用:

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

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

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

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

簡素化

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

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