derivative f(x)=(2x)(4x^2+6x^7+8)

解

導関数

f (x) = (2 x) (4 x^{2} + 6 x^{7} + 8)

2 (48 x^{7} + 12 x^{2} + 8)

解答ステップ

\frac{d}{d x} (2 x (4 x^{2} + 6 x^{7} + 8))

定数を除去:

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

= 2 \frac{d}{d x} (x (4 x^{2} + 6 x^{7} + 8))

積の法則を適用:

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

f = x, g = 4 x^{2} + 6 x^{7} + 8

= 2 (\frac{d x}{d x} (4 x^{2} + 6 x^{7} + 8) + \frac{d}{d x} (4 x^{2} + 6 x^{7} + 8) x)

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

\frac{d}{d x} (4 x^{2} + 6 x^{7} + 8) = 42 x^{6} + 8 x

= 2 (1 \cdot (4 x^{2} + 6 x^{7} + 8) + (42 x^{6} + 8 x) x)

簡素化

2 (1 \cdot (4 x^{2} + 6 x^{7} + 8) + (42 x^{6} + 8 x) x) : 2 (48 x^{7} + 12 x^{2} + 8)

= 2 (48 x^{7} + 12 x^{2} + 8)