derivative of 4xy^2-xy^3-x^2y^2

解

\frac{d}{d x} (4 x y^{2} - x y^{3} - x^{2} y^{2})

4 y^{2} + 8 x y \frac{d y}{d x} - y^{3} - 3 x y^{2} \frac{d y}{d x} - 2 x y^{2} - 2 x^{2} y \frac{d y}{d x}

解答ステップ

\frac{d}{d x} (4 x y^{2} - x y^{3} - x^{2} y^{2})

和/差の法則を適用:

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

= \frac{d}{d x} (4 x y^{2}) - \frac{d}{d x} (x y^{3}) - \frac{d}{d x} (x^{2} y^{2})

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

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

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

= 4 (y^{2} + 2 x y \frac{d y}{d x}) - (y^{3} + 3 x y^{2} \frac{d y}{d x}) - (2 x y^{2} + 2 y \frac{d y}{d x} x^{2})

簡素化

4 (y^{2} + 2 x y \frac{d y}{d x}) - (y^{3} + 3 x y^{2} \frac{d y}{d x}) - (2 x y^{2} + 2 y \frac{d y}{d x} x^{2}) : 4 y^{2} + 8 x y \frac{d y}{d x} - y^{3} - 3 x y^{2} \frac{d y}{d x} - 2 x y^{2} - 2 x^{2} y \frac{d y}{d x}

= 4 y^{2} + 8 x y \frac{d y}{d x} - y^{3} - 3 x y^{2} \frac{d y}{d x} - 2 x y^{2} - 2 x^{2} y \frac{d y}{d x}