derivative of (csc(x)/x)

解

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

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

解答ステップ

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

商の法則を適用:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

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

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

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

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

簡素化

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