derivative f(x)=8sin^3(sqrt(x))

解

導関数

f (x) = 8 sin^{3} (x)

\frac{12 sin ^{2} ( x ) cos ( x )}{x}

解答ステップ

\frac{d}{d x} (8 sin^{3} (x))

定数を除去:

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

= 8 \frac{d}{d x} (sin^{3} (x))

連鎖法則を適用:

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

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

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

= 8 \cdot 3 (sin (x))^{2} \frac{cos ( x )}{2 x}

簡素化

8 \cdot 3 sin^{2} (x) \frac{cos ( x )}{2 x} : \frac{12 sin ^{2} ( x ) cos ( x )}{x}

= \frac{12 sin ^{2} ( x ) cos ( x )}{x}