derivative of xy^2-x^3y

解

\frac{d}{d x} (x y^{2} - x^{3} y)

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

解答ステップ

\frac{d}{d x} (x y^{2} - x^{3} y)

和/差の法則を適用:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d x} (x y^{2}) - \frac{d}{d x} (x^{3} y)

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

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

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

簡素化

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

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