derivative y=x^2sqrt(9-x^2)

解

導関数

y = x^{2} 9 - x^{2}

\frac{- 3 x ^{3} + 18 x}{9 - x ^{2}}

解答ステップ

\frac{d}{d x} (x^{2} 9 - x^{2})

積の法則を適用:

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

f = x^{2}, g = 9 - x^{2}

= \frac{d}{d x} (x^{2}) 9 - x^{2} + \frac{d}{d x} (9 - x^{2}) x^{2}

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

\frac{d}{d x} (9 - x^{2}) = - \frac{x}{9 - x ^{2}}

= 2 x 9 - x^{2} + (- \frac{x}{9 - x ^{2}}) x^{2}

簡素化

2 x 9 - x^{2} + (- \frac{x}{9 - x ^{2}}) x^{2} : \frac{- 3 x ^{3} + 18 x}{9 - x ^{2}}

= \frac{- 3 x ^{3} + 18 x}{9 - x ^{2}}