derivative of (cos(x-cos(3x))/(sin(2x)))

解

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

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

解答ステップ

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

\frac{d}{d x} (cos (x) - cos (3 x)) = - sin (x) + 3 sin (3 x)

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

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

簡素化

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