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

解

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

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

解答ステップ

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

積の法則を適用:

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

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

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

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

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

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

簡素化

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

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