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

解

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

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

解答ステップ

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

\frac{d x}{d x} = 1

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

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

簡素化

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

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