derivative of (2-sec(x)/(tan(x)))

解

\frac{d}{d x} (\frac{2 - sec ( x )}{tan ( x )})

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

解答ステップ

\frac{d}{d x} (\frac{2 - sec ( x )}{tan ( 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 - sec ( x ) ) tan ( x ) - \frac{d}{d x} ( tan ( x ) ) ( 2 - sec ( x ) )}{( tan ( x ) ) ^{2}}

\frac{d}{d x} (2 - sec (x)) = - sec (x) tan (x)

\frac{d}{d x} (tan (x)) = sec^{2} (x)

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

簡素化

\frac{( - sec ( x ) tan ( x ) ) tan ( x ) - sec ^{2} ( x ) ( 2 - sec ( x ) )}{tan ^{2} ( x )} : \frac{- tan ^{2} ( x ) sec ( x ) - sec ^{2} ( x ) ( 2 - sec ( x ) )}{tan ^{2} ( x )}

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