derivative of (x(2x^2y+4))

解

\frac{d}{d x} ((x) (2 x^{2} y + 4))

2 x^{3} \frac{d y}{d x} + 6 x^{2} y + 4

解答ステップ

\frac{d}{d x} ((x) (2 x^{2} y + 4))

積の法則を適用:

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

f = x, g = 2 x^{2} y + 4

= \frac{d x}{d x} (2 x^{2} y + 4) + \frac{d}{d x} (2 x^{2} y + 4) x

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

\frac{d}{d x} (2 x^{2} y + 4) = 2 (x^{2} \frac{d y}{d x} + 2 x y)

= 1 \cdot (2 x^{2} y + 4) + 2 (x^{2} \frac{d y}{d x} + 2 x y) x

簡素化

1 \cdot (2 x^{2} y + 4) + 2 (x^{2} \frac{d y}{d x} + 2 x y) x : 2 x^{3} \frac{d y}{d x} + 6 x^{2} y + 4

= 2 x^{3} \frac{d y}{d x} + 6 x^{2} y + 4