derivative of (2x/(sin(x)))

解

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

2 (csc (x) - x cot (x) csc (x))

解答ステップ

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

簡素化

\frac{2 x}{sin ( x )} : 2 x csc (x)

= \frac{d}{d x} (2 x csc (x))

定数を除去:

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

= 2 \frac{d}{d x} (x csc (x))

積の法則を適用:

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

f = x, g = csc (x)

= 2 (\frac{d x}{d x} csc (x) + \frac{d}{d x} (csc (x)) x)

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

\frac{d}{d x} (csc (x)) = - cot (x) csc (x)

= 2 (1 \cdot csc (x) + (- cot (x) csc (x)) x)

簡素化

= 2 (csc (x) - x cot (x) csc (x))