derivative 6sin^5(x)cos(x)

解

導関数

6 sin^{5} (x) cos (x)

6 (5 sin^{4} (x) cos^{2} (x) - sin^{6} (x))

解答ステップ

\frac{d}{d x} (6 sin^{5} (x) cos (x))

定数を除去:

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

= 6 \frac{d}{d x} (sin^{5} (x) cos (x))

積の法則を適用:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = sin^{5} (x), g = cos (x)

= 6 (\frac{d}{d x} (sin^{5} (x)) cos (x) + \frac{d}{d x} (cos (x)) sin^{5} (x))

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

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

= 6 (5 sin^{4} (x) cos (x) cos (x) + (- sin (x)) sin^{5} (x))

簡素化

6 (5 sin^{4} (x) cos (x) cos (x) + (- sin (x)) sin^{5} (x)) : 6 (5 sin^{4} (x) cos^{2} (x) - sin^{6} (x))

= 6 (5 sin^{4} (x) cos^{2} (x) - sin^{6} (x))