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

解

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

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

解答ステップ

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

簡素化

\frac{cos ( x )}{x} + \frac{x}{cos ( x )} : \frac{cos ( x )}{x} + x sec (x)

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

和/差の法則を適用:

(f \pm g)^{'} = f^{'} \pm g^{'}

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

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

\frac{d}{d x} (x sec (x)) = sec (x) + x sec (x) tan (x)

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