derivative of 1/(sqrt(cos(x)))

解

\frac{d}{d x} (\frac{1}{cos ( x )})

\frac{sin ( x )}{2 cos ^{\frac{3}{2}} ( x )}

解答ステップ

\frac{d}{d x} (\frac{1}{cos ( x )})

指数の規則を適用する:

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

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

連鎖法則を適用:

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

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

\frac{d}{d x} (cos (x)) = - sin (x)

= - \frac{1}{2 ( cos ( x ) ) ^{\frac{3}{2}}} (- sin (x))

簡素化

- \frac{1}{2 cos ^{\frac{3}{2}} ( x )} (- sin (x)) : \frac{sin ( x )}{2 cos ^{\frac{3}{2}} ( x )}

= \frac{sin ( x )}{2 cos ^{\frac{3}{2}} ( x )}