derivative of (cos(2x)/(sin(4x)))

解

\frac{d}{d x} (\frac{cos ( 2 x )}{sin ( 4 x )})

- 2 sin (2 x) csc (4 x) - 4 cot (4 x) csc (4 x) cos (2 x)

解答ステップ

\frac{d}{d x} (\frac{cos ( 2 x )}{sin ( 4 x )})

簡素化

\frac{cos ( 2 x )}{sin ( 4 x )} : cos (2 x) csc (4 x)

= \frac{d}{d x} (cos (2 x) csc (4 x))

積の法則を適用:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = cos (2 x), g = csc (4 x)

= \frac{d}{d x} (cos (2 x)) csc (4 x) + \frac{d}{d x} (csc (4 x)) cos (2 x)

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

\frac{d}{d x} (csc (4 x)) = - cot (4 x) csc (4 x) \cdot 4

= (- sin (2 x) \cdot 2) csc (4 x) + (- cot (4 x) csc (4 x) \cdot 4) cos (2 x)

簡素化

= - 2 sin (2 x) csc (4 x) - 4 cot (4 x) csc (4 x) cos (2 x)