derivative f(x)=xsqrt(8-x^2)

解

導関数

f (x) = x 8 - x^{2}

\frac{- 2 x ^{2} + 8}{8 - x ^{2}}

解答ステップ

\frac{d}{d x} (x 8 - x^{2})

積の法則を適用:

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

f = x, g = 8 - x^{2}

= \frac{d x}{d x} 8 - x^{2} + \frac{d}{d x} (8 - x^{2}) x

\frac{d x}{d x} = 1

\frac{d}{d x} (8 - x^{2}) = - \frac{x}{8 - x ^{2}}

= 1 \cdot 8 - x^{2} + (- \frac{x}{8 - x ^{2}}) x

簡素化

1 \cdot 8 - x^{2} + (- \frac{x}{8 - x ^{2}}) x : \frac{- 2 x ^{2} + 8}{8 - x ^{2}}

= \frac{- 2 x ^{2} + 8}{8 - x ^{2}}