derivative f(x)=(8x(x^2+6)^3)

解

導関数

f (x) = (8 x (x^{2} + 6)^{3})

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

解答ステップ

\frac{d}{d x} (8 x (x^{2} + 6)^{3})

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

= 8 \frac{d}{d x} (x (x^{2} + 6)^{3})

積の法則を適用:

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

f = x, g = (x^{2} + 6)^{3}

= 8 (\frac{d x}{d x} (x^{2} + 6)^{3} + \frac{d}{d x} ((x^{2} + 6)^{3}) x)

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

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

= 8 (1 \cdot (x^{2} + 6)^{3} + 6 x (x^{2} + 6)^{2} x)

簡素化

8 (1 \cdot (x^{2} + 6)^{3} + 6 x (x^{2} + 6)^{2} x) : 8 ((x^{2} + 6)^{3} + 6 x^{2} (x^{2} + 6)^{2})

= 8 ((x^{2} + 6)^{3} + 6 x^{2} (x^{2} + 6)^{2})