derivative of x(x^2+1^3)

解

\frac{d}{d x} (x (x^{2} + 1)^{3})

(x^{2} + 1)^{3} + 6 x^{2} (x^{2} + 1)^{2}

解答ステップ

\frac{d}{d x} (x (x^{2} + 1)^{3})

積の法則を適用:

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

f = x, g = (x^{2} + 1)^{3}

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

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

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

= 1 \cdot (x^{2} + 1)^{3} + 6 x (x^{2} + 1)^{2} x

簡素化

1 \cdot (x^{2} + 1)^{3} + 6 x (x^{2} + 1)^{2} x : (x^{2} + 1)^{3} + 6 x^{2} (x^{2} + 1)^{2}

= (x^{2} + 1)^{3} + 6 x^{2} (x^{2} + 1)^{2}