derivative y=(x^2-6x+10)sqrt(25-x^2)

解

導関数

y = (x^{2} - 6 x + 10) 25 - x^{2}

\frac{- 3 x ^{3} + 12 x ^{2} + 40 x - 150}{25 - x ^{2}}

解答ステップ

\frac{d}{d x} ((x^{2} - 6 x + 10) 25 - x^{2})

積の法則を適用:

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

f = x^{2} - 6 x + 10, g = 25 - x^{2}

= \frac{d}{d x} (x^{2} - 6 x + 10) 25 - x^{2} + \frac{d}{d x} (25 - x^{2}) (x^{2} - 6 x + 10)

\frac{d}{d x} (x^{2} - 6 x + 10) = 2 x - 6

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

= (2 x - 6) 25 - x^{2} + (- \frac{x}{25 - x ^{2}}) (x^{2} - 6 x + 10)

簡素化

(2 x - 6) 25 - x^{2} + (- \frac{x}{25 - x ^{2}}) (x^{2} - 6 x + 10) : \frac{- 3 x ^{3} + 12 x ^{2} + 40 x - 150}{25 - x ^{2}}

= \frac{- 3 x ^{3} + 12 x ^{2} + 40 x - 150}{25 - x ^{2}}