derivative of 4/(sqrt(sec(x)))

解

\frac{d}{d x} (\frac{4}{sec ( x )})

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

解答ステップ

\frac{d}{d x} (\frac{4}{sec ( x )})

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

= 4 \frac{d}{d x} (\frac{1}{sec ( x )})

指数の規則を適用する:

\frac{1}{a} = a^{- 1}

= 4 \frac{d}{d x} (sec^{- \frac{1}{2}} (x))

連鎖法則を適用:

- \frac{1}{2 ( sec ( x ) ) ^{\frac{3}{2}}} \frac{d}{d x} (sec (x))

= - \frac{1}{2 ( sec ( x ) ) ^{\frac{3}{2}}} \frac{d}{d x} (sec (x))

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

= 4 (- \frac{1}{2 ( sec ( x ) ) ^{\frac{3}{2}}} sec (x) tan (x))

簡素化

4 (- \frac{1}{2 sec ^{\frac{3}{2}} ( x )} sec (x) tan (x)) : - 2 tan (x) cos^{\frac{1}{2}} (x)

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