derivative (x^2-2x+5)sqrt(25-x^2)

解答

导数

(x^{2} - 2 x + 5) 25 - x^{2}

\frac{- 3 x ^{3} + 4 x ^{2} + 45 x - 50}{25 - x ^{2}}

求解步骤

\frac{d}{d x} ((x^{2} - 2 x + 5) 25 - x^{2})

使用微分乘法定则:

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

f = x^{2} - 2 x + 5, g = 25 - x^{2}

= \frac{d}{d x} (x^{2} - 2 x + 5) 25 - x^{2} + \frac{d}{d x} (25 - x^{2}) (x^{2} - 2 x + 5)

\frac{d}{d x} (x^{2} - 2 x + 5) = 2 x - 2

\frac{d}{d x} (25 - x^{2}) = - \frac{x}{25 - x ^{2}}

= (2 x - 2) 25 - x^{2} + (- \frac{x}{25 - x ^{2}}) (x^{2} - 2 x + 5)

化简

(2 x - 2) 25 - x^{2} + (- \frac{x}{25 - x ^{2}}) (x^{2} - 2 x + 5) : \frac{- 3 x ^{3} + 4 x ^{2} + 45 x - 50}{25 - x ^{2}}

= \frac{- 3 x ^{3} + 4 x ^{2} + 45 x - 50}{25 - x ^{2}}

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