derivative of (1+sec(x)/(1-sec(x)))

解

\frac{d}{d x} (\frac{1 + sec ( x )}{1 - sec ( x )})

\frac{2 sec ( x ) tan ( x )}{( 1 - sec ( x ) ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{1 + sec ( x )}{1 - sec ( 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} ( 1 + sec ( x ) ) ( 1 - sec ( x ) ) - \frac{d}{d x} ( 1 - sec ( x ) ) ( 1 + sec ( x ) )}{( 1 - sec ( x ) ) ^{2}}

\frac{d}{d x} (1 + sec (x)) = sec (x) tan (x)

\frac{d}{d x} (1 - sec (x)) = - sec (x) tan (x)

= \frac{sec ( x ) tan ( x ) ( 1 - sec ( x ) ) - ( - sec ( x ) tan ( x ) ) ( 1 + sec ( x ) )}{( 1 - sec ( x ) ) ^{2}}

簡素化

\frac{sec ( x ) tan ( x ) ( 1 - sec ( x ) ) - ( - sec ( x ) tan ( x ) ) ( 1 + sec ( x ) )}{( 1 - sec ( x ) ) ^{2}} : \frac{2 sec ( x ) tan ( x )}{( 1 - sec ( x ) ) ^{2}}

= \frac{2 sec ( x ) tan ( x )}{( 1 - sec ( x ) ) ^{2}}