derivative of (x+y/(2+cos(x)))

解

\frac{d}{d x} (\frac{x + y}{2 + cos ( x )})

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

解答ステップ

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

\frac{d}{d x} (x + y) = 1 + \frac{d y}{d x}

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

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

簡素化

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