derivative of (sin(x)/(3x))

解

\frac{d}{d x} (\frac{sin ( x )}{3 x})

\frac{x cos ( x ) - sin ( x )}{3 x ^{2}}

解答ステップ

\frac{d}{d x} (\frac{sin ( x )}{3 x})

定数を除去:

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

= \frac{1}{3} \frac{d}{d x} (\frac{sin ( x )}{x})

商の法則を適用:

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

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

\frac{d}{d x} (sin (x)) = cos (x)

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

= \frac{1}{3} \cdot \frac{cos ( x ) x - 1 \cdot sin ( x )}{x ^{2}}

簡素化

\frac{1}{3} \cdot \frac{cos ( x ) x - 1 \cdot sin ( x )}{x ^{2}} : \frac{x cos ( x ) - sin ( x )}{3 x ^{2}}

= \frac{x cos ( x ) - sin ( x )}{3 x ^{2}}