derivative of (cos(x)/(cos(x)+1))

解答

\frac{d}{d x} (\frac{cos ( x )}{cos ( x ) + 1})

- \frac{sin ( x )}{( cos ( x ) + 1 ) ^{2}}

求解步骤

\frac{d}{d x} (\frac{cos ( x )}{cos ( x ) + 1})

使用微分除法定则:

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

= \frac{\frac{d}{d x} ( cos ( x ) ) ( cos ( x ) + 1 ) - \frac{d}{d x} ( cos ( x ) + 1 ) cos ( x )}{( cos ( x ) + 1 ) ^{2}}

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

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

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

化简

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

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

s im pl i f y \frac{x ^{2} + x + 1}{2 x}

\int \frac{8 x}{8 x + 3} d x

\frac{d}{d x} ((7 - 3 x^{3})^{2})

\int_{0}^{2} x e^{- x} d x

x \to 0 lim ((x - 2) e^{- \frac{1}{x}})