derivative of 2arctan(sqrt(sin(x)))

解

\frac{d}{d x} (2 arctan (sin (x)))

\frac{cos ( x )}{sin ( x ) ( sin ( x ) + 1 )}

解答ステップ

\frac{d}{d x} (2 arctan (sin (x)))

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

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

連鎖法則を適用:

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

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

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

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

簡素化

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

= \frac{cos ( x )}{sin ( x ) ( sin ( x ) + 1 )}