derivative of xsqrt(9-x^2)

解答

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

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

求解步骤

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

使用微分乘法定则:

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

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

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

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

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

= 1 \cdot 9 - x^{2} + (- \frac{x}{9 - x ^{2}}) x

化简

1 \cdot 9 - x^{2} + (- \frac{x}{9 - x ^{2}}) x : \frac{- 2 x ^{2} + 9}{9 - x ^{2}}

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

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