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

解答

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

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

求解步骤

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

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

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

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

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

化简

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

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

\int \frac{6}{1 + x ^{2}} d x

x \to 1 lim (f (x) + g (x))

n = 2 \sum \infty \frac{n}{2 n ^{2} - 3}

\frac{\partial}{\partial x} (e^{- x y} + y^{3} x^{4})

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