derivative of sin^{-2}(x^2+x)

解

\frac{d}{d x} (sin^{- 2} (x^{2} + x))

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

解答ステップ

\frac{d}{d x} (sin^{- 2} (x^{2} + x))

連鎖法則を適用:

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

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

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

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

分数を乗じる:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

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