derivative of (1+sin(x)/(xcos(x)))

解

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

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

解答ステップ

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

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

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

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

簡素化

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

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