derivative f(x)=(sin(4x))/(sin(5x))

解

導関数

f (x) = \frac{sin ( 4 x )}{sin ( 5 x )}

4 cos (4 x) csc (5 x) - 5 cot (5 x) csc (5 x) sin (4 x)

解答ステップ

\frac{d}{d x} (\frac{sin ( 4 x )}{sin ( 5 x )})

簡素化

\frac{sin ( 4 x )}{sin ( 5 x )} : sin (4 x) csc (5 x)

= \frac{d}{d x} (sin (4 x) csc (5 x))

積の法則を適用:

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

f = sin (4 x), g = csc (5 x)

= \frac{d}{d x} (sin (4 x)) csc (5 x) + \frac{d}{d x} (csc (5 x)) sin (4 x)

\frac{d}{d x} (sin (4 x)) = cos (4 x) \cdot 4

\frac{d}{d x} (csc (5 x)) = - cot (5 x) csc (5 x) \cdot 5

= cos (4 x) \cdot 4 csc (5 x) + (- cot (5 x) csc (5 x) \cdot 5) sin (4 x)

簡素化

= 4 cos (4 x) csc (5 x) - 5 cot (5 x) csc (5 x) sin (4 x)