derivative of ((2pi-xcos(x))/(x-2pi))

解

\frac{d}{d x} (\frac{( 2 π - x cos ( x ) )}{x - 2 π})

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

解答ステップ

\frac{d}{d x} (\frac{( 2 π - x cos ( x ) )}{x - 2 π})

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

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

\frac{d}{d x} (x - 2 π) = 1

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

簡素化

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

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