derivative of 4sin^5(sqrt(x))

解

\frac{d}{d x} (4 sin^{5} (x))

\frac{10 sin ^{4} ( x ) cos ( x )}{x}

解答ステップ

\frac{d}{d x} (4 sin^{5} (x))

定数を除去:

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

= 4 \frac{d}{d x} (sin^{5} (x))

連鎖法則を適用:

5 (sin (x))^{4} \frac{d}{d x} (sin (x))

= 5 (sin (x))^{4} \frac{d}{d x} (sin (x))

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

= 4 \cdot 5 (sin (x))^{4} \frac{cos ( x )}{2 x}

簡素化

4 \cdot 5 sin^{4} (x) \frac{cos ( x )}{2 x} : \frac{10 sin ^{4} ( x ) cos ( x )}{x}

= \frac{10 sin ^{4} ( x ) cos ( x )}{x}