derivative of a(1-cos^2(x/2)^2)

解

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

2 a sin^{3} (\frac{x}{2}) cos (\frac{x}{2})

解答ステップ

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

簡素化

a (1 - cos^{2} (\frac{x}{2}))^{2} : a sin^{4} (\frac{x}{2})

= \frac{d}{d x} (a sin^{4} (\frac{x}{2}))

定数を除去:

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

= a \frac{d}{d x} (sin^{4} (\frac{x}{2}))

連鎖法則を適用:

4 (sin (\frac{x}{2}))^{3} \frac{d}{d x} (sin (\frac{x}{2}))

= 4 (sin (\frac{x}{2}))^{3} \frac{d}{d x} (sin (\frac{x}{2}))

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

= a \cdot 4 (sin (\frac{x}{2}))^{3} cos (\frac{x}{2}) \frac{1}{2}

簡素化

a \cdot 4 sin^{3} (\frac{x}{2}) cos (\frac{x}{2}) \frac{1}{2} : 2 a sin^{3} (\frac{x}{2}) cos (\frac{x}{2})

= 2 a sin^{3} (\frac{x}{2}) cos (\frac{x}{2})