derivative f(θ)=(sec(θ))/(1+sec(θ))

解

導関数

f (θ) = \frac{sec ( θ )}{1 + sec ( θ )}

\frac{sin ( θ )}{( 1 + cos ( θ ) ) ^{2}}

解答ステップ

\frac{d}{d θ} (\frac{sec ( θ )}{1 + sec ( θ )})

商の法則を適用:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

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

\frac{d}{d θ} (sec (θ)) = sec (θ) tan (θ)

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

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

簡素化

\frac{sec ( θ ) tan ( θ ) ( 1 + sec ( θ ) ) - sec ( θ ) tan ( θ ) sec ( θ )}{( 1 + sec ( θ ) ) ^{2}} : \frac{sin ( θ )}{( 1 + cos ( θ ) ) ^{2}}

= \frac{sin ( θ )}{( 1 + cos ( θ ) ) ^{2}}