derivative f(x)=(cos(x))/(1+asin(x))

解

導関数

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

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

解答ステップ

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

商の法則を適用:

(\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}}

簡素化

\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}}