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

解

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

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

解答ステップ

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

定数を除去:

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

= \frac{1}{2} \frac{d}{d x} (\frac{sin ( x ) cos ( x )}{x ^{2}})

商の法則を適用:

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

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

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

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

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

簡素化

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

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