derivative (sec(9x))/(1+sec(9x))

解

導関数

\frac{sec ( 9 x )}{1 + sec ( 9 x )}

\frac{9 sec ( 9 x ) tan ( 9 x )}{( 1 + sec ( 9 x ) ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{sec ( 9 x )}{1 + sec ( 9 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} ( sec ( 9 x ) ) ( 1 + sec ( 9 x ) ) - \frac{d}{d x} ( 1 + sec ( 9 x ) ) sec ( 9 x )}{( 1 + sec ( 9 x ) ) ^{2}}

\frac{d}{d x} (sec (9 x)) = sec (9 x) tan (9 x) \cdot 9

\frac{d}{d x} (1 + sec (9 x)) = 9 sec (9 x) tan (9 x)

= \frac{sec ( 9 x ) tan ( 9 x ) \cdot 9 ( 1 + sec ( 9 x ) ) - 9 sec ( 9 x ) tan ( 9 x ) sec ( 9 x )}{( 1 + sec ( 9 x ) ) ^{2}}

簡素化

\frac{sec ( 9 x ) tan ( 9 x ) \cdot 9 ( 1 + sec ( 9 x ) ) - 9 sec ( 9 x ) tan ( 9 x ) sec ( 9 x )}{( 1 + sec ( 9 x ) ) ^{2}} : \frac{9 sec ( 9 x ) tan ( 9 x )}{( 1 + sec ( 9 x ) ) ^{2}}

= \frac{9 sec ( 9 x ) tan ( 9 x )}{( 1 + sec ( 9 x ) ) ^{2}}