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

解

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

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

解答ステップ

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

積の法則を適用:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = \frac{x + π}{x}, g = cos (x)

= \frac{d}{d x} (\frac{x + π}{x}) cos (x) + \frac{d}{d x} (cos (x)) \frac{x + π}{x}

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

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

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

簡素化

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

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