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

解

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

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

解答ステップ

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

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

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

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

簡素化

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

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