derivative f(x)=(cos(x)-4)/(2sin(x))

解答

导数

f (x) = \frac{cos ( x ) - 4}{2 sin ( x )}

\frac{- sin ^{2} ( x ) - cos ( x ) ( cos ( x ) - 4 )}{2 sin ^{2} ( x )}

求解步骤

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

将常数提出:

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

= \frac{1}{2} \frac{d}{d x} (\frac{cos ( x ) - 4}{sin ( x )})

使用微分除法定则:

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

= \frac{1}{2} \cdot \frac{\frac{d}{d x} ( cos ( x ) - 4 ) sin ( x ) - \frac{d}{d x} ( sin ( x ) ) ( cos ( x ) - 4 )}{( sin ( x ) ) ^{2}}

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

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

= \frac{1}{2} \cdot \frac{( - sin ( x ) ) sin ( x ) - cos ( x ) ( cos ( x ) - 4 )}{( sin ( x ) ) ^{2}}

化简

\frac{1}{2} \cdot \frac{( - sin ( x ) ) sin ( x ) - cos ( x ) ( cos ( x ) - 4 )}{sin ^{2} ( x )} : \frac{- sin ^{2} ( x ) - cos ( x ) ( cos ( x ) - 4 )}{2 sin ^{2} ( x )}

= \frac{- sin ^{2} ( x ) - cos ( x ) ( cos ( x ) - 4 )}{2 sin ^{2} ( x )}

\frac{d}{d x} ((x^{4} + 3 x^{2}) (1 - 5 x))

\int e^{- b x} d x

\int_{1}^{3} x^{2} ln (x) d x

\int \frac{5 x ^{2} + 2 x - 5}{x ^{3} - x} d x

\frac{\partial}{\partial x} (6 y - 5 x^{2} e^{5 y})