derivative of arcsin(sqrt(sin(x)))

解

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

\frac{cos ( x )}{2 sin ( x ) 1 - sin ( x )}

解答ステップ

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

連鎖法則を適用:

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

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

\frac{d}{d x} (sin (x)) = \frac{cos ( x )}{2 sin ( x )}

= \frac{1}{1 - ( sin ( x ) ) ^{2}} \cdot \frac{cos ( x )}{2 sin ( x )}

簡素化

\frac{1}{1 - ( sin ( x ) ) ^{2}} \cdot \frac{cos ( x )}{2 sin ( x )} : \frac{cos ( x )}{2 sin ( x ) 1 - sin ( x )}

= \frac{cos ( x )}{2 sin ( x ) 1 - sin ( x )}