derivative of sin^n(xcos(n)x)

解

\frac{d}{d x} (sin^{n} (x) cos (n) x)

cos (n) sin^{n} (x) (n x cot (x) + 1)

解答ステップ

\frac{d}{d x} (sin^{n} (x) cos (n) x)

定数を除去:

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

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

積の法則を適用:

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

f = sin^{n} (x), g = x

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

\frac{d}{d x} (sin^{n} (x)) = n cos (x) sin^{n - 1} (x)

\frac{d x}{d x} = 1

= cos (n) (n cos (x) sin^{n - 1} (x) x + 1 \cdot sin^{n} (x))

簡素化

cos (n) (n cos (x) sin^{n - 1} (x) x + 1 \cdot sin^{n} (x)) : cos (n) sin^{n} (x) (n x cot (x) + 1)

= cos (n) sin^{n} (x) (n x cot (x) + 1)