derivative of 1/(arccos(x^2+2x+4))

解

\frac{d}{d x} (\frac{1}{arccos ( x ^{2} + 2 x + 4 )})

\frac{2 x + 2}{arccos ^{2} ( x ^{2} + 2 x + 4 ) - x ^{4} - 4 x ^{3} - 12 x ^{2} - 16 x - 15}

解答ステップ

\frac{d}{d x} (\frac{1}{arccos ( x ^{2} + 2 x + 4 )})

指数の規則を適用する:

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

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

連鎖法則を適用:

- \frac{1}{( arccos ( x ^{2} + 2 x + 4 ) ) ^{2}} \frac{d}{d x} (arccos (x^{2} + 2 x + 4))

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

\frac{d}{d x} (arccos (x^{2} + 2 x + 4)) = - \frac{2 x + 2}{- x ^{4} - 4 x ^{3} - 12 x ^{2} - 16 x - 15}

= - \frac{1}{( arccos ( x ^{2} + 2 x + 4 ) ) ^{2}} (- \frac{2 x + 2}{- x ^{4} - 4 x ^{3} - 12 x ^{2} - 16 x - 15})

簡素化

- \frac{1}{arccos ^{2} ( x ^{2} + 2 x + 4 )} (- \frac{2 x + 2}{- x ^{4} - 4 x ^{3} - 12 x ^{2} - 16 x - 15}) : \frac{2 x + 2}{arccos ^{2} ( x ^{2} + 2 x + 4 ) - x ^{4} - 4 x ^{3} - 12 x ^{2} - 16 x - 15}

= \frac{2 x + 2}{arccos ^{2} ( x ^{2} + 2 x + 4 ) - x ^{4} - 4 x ^{3} - 12 x ^{2} - 16 x - 15}