derivative of x^2*(sin(x)^{-1})

解

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

\frac{2 x - x ^{2} cot ( x )}{sin ( x )}

解答ステップ

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

積の法則を適用:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x^{2}, g = (sin (x))^{- 1}

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

\frac{d}{d x} (x^{2}) = 2 x

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

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

簡素化

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

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