derivative of (cot(x)/(sin(x)))

解

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

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

解答ステップ

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

簡素化

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

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

積の法則を適用:

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

f = cot (x), g = csc (x)

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

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

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

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

簡素化

(- csc^{2} (x)) csc (x) + (- cot (x) csc (x)) cot (x) : csc (x) - 2 csc^{3} (x)

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