derivative of (x^{2-x}^5(x-3x^2))

解

\frac{d}{d x} ((x^{2 - x})^{5} (x - 3 x^{2}))

15 x^{- 5 x + 12} ln (x) + 15 x^{- 5 x + 12} - 5 x^{- 5 x + 11} ln (x) - 41 x^{- 5 x + 11} + 10 x^{- 5 x + 10} + x^{5 (- x + 2)}

解答ステップ

\frac{d}{d x} ((x^{2 - x})^{5} (x - 3 x^{2}))

積の法則を適用:

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

f = (x^{2 - x})^{5}, g = x - 3 x^{2}

= \frac{d}{d x} ((x^{2 - x})^{5}) (x - 3 x^{2}) + \frac{d}{d x} (x - 3 x^{2}) (x^{2 - x})^{5}

\frac{d}{d x} ((x^{2 - x})^{5}) = 5 x^{- 5 x + 10} (- ln (x) + \frac{2 - x}{x})

\frac{d}{d x} (x - 3 x^{2}) = 1 - 6 x

= 5 x^{- 5 x + 10} (- ln (x) + \frac{2 - x}{x}) (x - 3 x^{2}) + (1 - 6 x) (x^{2 - x})^{5}

簡素化

5 x^{- 5 x + 10} (- ln (x) + \frac{2 - x}{x}) (x - 3 x^{2}) + (1 - 6 x) (x^{2 - x})^{5} : 15 x^{- 5 x + 12} ln (x) + 15 x^{- 5 x + 12} - 5 x^{- 5 x + 11} ln (x) - 41 x^{- 5 x + 11} + 10 x^{- 5 x + 10} + x^{5 (- x + 2)}

= 15 x^{- 5 x + 12} ln (x) + 15 x^{- 5 x + 12} - 5 x^{- 5 x + 11} ln (x) - 41 x^{- 5 x + 11} + 10 x^{- 5 x + 10} + x^{5 (- x + 2)}