derivative of x/(tan(3x-1))

解

\frac{d}{d x} (\frac{x}{tan ( 3 x - 1 )})

cot (3 x - 1) - 3 x csc^{2} (3 x - 1)

解答ステップ

\frac{d}{d x} (\frac{x}{tan ( 3 x - 1 )})

簡素化

\frac{x}{tan ( 3 x - 1 )} : x cot (3 x - 1)

= \frac{d}{d x} (x cot (3 x - 1))

積の法則を適用:

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

f = x, g = cot (3 x - 1)

= \frac{d x}{d x} cot (3 x - 1) + \frac{d}{d x} (cot (3 x - 1)) x

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

\frac{d}{d x} (cot (3 x - 1)) = - csc^{2} (3 x - 1) \cdot 3

= 1 \cdot cot (3 x - 1) + (- csc^{2} (3 x - 1) \cdot 3) x

簡素化

= cot (3 x - 1) - 3 x csc^{2} (3 x - 1)