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

解答

导数

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)

求解步骤

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

化简

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

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

使用微分乘法定则:

(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)

化简

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

d er i v a t i v e y = arctan (3 x^{2})

\int e^{10 t} d t

\frac{d}{d x} (f (x)^{- 1} (x))

x^{2} y^{^{'}} + 2 x y = 21 cos^{2} (x)

d er i v a t i v e \frac{x}{x - 1}