derivative y=(x^2+1)^5(1-x^2)^5

解

導関数

y = (x^{2} + 1)^{5} (1 - x^{2})^{5}

10 x (x^{2} + 1)^{4} (1 - x^{2})^{5} - 10 x (1 - x^{2})^{4} (x^{2} + 1)^{5}

解答ステップ

\frac{d}{d x} ((x^{2} + 1)^{5} (1 - x^{2})^{5})

積の法則を適用:

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

f = (x^{2} + 1)^{5}, g = (1 - x^{2})^{5}

= \frac{d}{d x} ((x^{2} + 1)^{5}) (1 - x^{2})^{5} + \frac{d}{d x} ((1 - x^{2})^{5}) (x^{2} + 1)^{5}

\frac{d}{d x} ((x^{2} + 1)^{5}) = 10 x (x^{2} + 1)^{4}

\frac{d}{d x} ((1 - x^{2})^{5}) = - 10 x (1 - x^{2})^{4}

= 10 x (x^{2} + 1)^{4} (1 - x^{2})^{5} + (- 10 x (1 - x^{2})^{4}) (x^{2} + 1)^{5}

簡素化

= 10 x (x^{2} + 1)^{4} (1 - x^{2})^{5} - 10 x (1 - x^{2})^{4} (x^{2} + 1)^{5}