derivative of ln(sqrt((1-sin(x)/(1+sin(x)))))

解

\frac{d}{d x} (ln (\frac{1 - sin ( x )}{1 + sin ( x )}))

- sec (x)

解答ステップ

\frac{d}{d x} (ln (\frac{1 - sin ( x )}{1 + sin ( x )}))

連鎖法則を適用:

\frac{1}{\frac{1 - s i n ( x )}{1 + s i n ( x )}} \frac{d}{d x} (\frac{1 - sin ( x )}{1 + sin ( x )})

= \frac{1}{\frac{1 - s i n ( x )}{1 + s i n ( x )}} \frac{d}{d x} (\frac{1 - sin ( x )}{1 + sin ( x )})

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

= \frac{1}{\frac{1 - s i n ( x )}{1 + s i n ( x )}} (- \frac{cos ( x )}{- sin ( x ) + 1 ( sin ( x ) + 1 ) ^{\frac{3}{2}}})

簡素化

\frac{1}{\frac{1 - s i n ( x )}{1 + s i n ( x )}} (- \frac{cos ( x )}{- sin ( x ) + 1 ( sin ( x ) + 1 ) ^{\frac{3}{2}}}) : - sec (x)

= - sec (x)