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

解

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

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

解答ステップ

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

指数の規則を適用する:

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

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

連鎖法則を適用:

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

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

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

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

簡素化

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

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