derivative of 1/(csc(x+cot(x)))

解

\frac{d}{d x} (\frac{1}{csc ( x ) + cot ( x )})

\frac{csc ( x )}{csc ( x ) + cot ( x )}

解答ステップ

\frac{d}{d x} (\frac{1}{csc ( x ) + cot ( x )})

指数の規則を適用する:

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

= \frac{d}{d x} ((csc (x) + cot (x))^{- 1})

連鎖法則を適用:

- \frac{1}{( csc ( x ) + cot ( x ) ) ^{2}} \frac{d}{d x} (csc (x) + cot (x))

= - \frac{1}{( csc ( x ) + cot ( x ) ) ^{2}} \frac{d}{d x} (csc (x) + cot (x))

\frac{d}{d x} (csc (x) + cot (x)) = - cot (x) csc (x) - csc^{2} (x)

= - \frac{1}{( csc ( x ) + cot ( x ) ) ^{2}} (- cot (x) csc (x) - csc^{2} (x))

簡素化

- \frac{1}{( csc ( x ) + cot ( x ) ) ^{2}} (- cot (x) csc (x) - csc^{2} (x)) : \frac{csc ( x )}{csc ( x ) + cot ( x )}

= \frac{csc ( x )}{csc ( x ) + cot ( x )}