derivative of csc^2(xcot(x))

解

\frac{d}{d x} (csc^{2} (x) cot (x))

- 2 csc^{2} (x) cot^{2} (x) - csc^{4} (x)

解答ステップ

\frac{d}{d x} (csc^{2} (x) cot (x))

積の法則を適用:

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

f = csc^{2} (x), g = cot (x)

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

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

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

= (- 2 csc^{2} (x) cot (x)) cot (x) + (- csc^{2} (x)) csc^{2} (x)

簡素化

(- 2 csc^{2} (x) cot (x)) cot (x) + (- csc^{2} (x)) csc^{2} (x) : - 2 csc^{2} (x) cot^{2} (x) - csc^{4} (x)

= - 2 csc^{2} (x) cot^{2} (x) - csc^{4} (x)