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

解

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

- \frac{x sin ( 2 x ) - sin ^{2} ( x )}{x ^{2}}

解答ステップ

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

簡素化

\frac{- sin ^{2} ( x )}{x} : - \frac{sin ^{2} ( x )}{x}

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

定数を除去:

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

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

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

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

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

簡素化

= - \frac{x sin ( 2 x ) - sin ^{2} ( x )}{x ^{2}}