derivative of (2-x(y-2)(3-x-y))

解

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

x^{2} \frac{d y}{d x} + 2 x y + 2 x y \frac{d y}{d x} - 4 x - 7 x \frac{d y}{d x} + y^{2} + 10 \frac{d y}{d x} - 7 y - 4 y \frac{d y}{d x} + 10

解答ステップ

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

積の法則を適用:

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

f = 2 - x, g = (y - 2) (3 - x - y)

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

\frac{d}{d x} (2 - x) = - 1

\frac{d}{d x} ((y - 2) (3 - x - y)) = - x \frac{d y}{d x} + 5 \frac{d y}{d x} - 2 y \frac{d y}{d x} - y + 2

= (- 1) (y - 2) (3 - x - y) + (- x \frac{d y}{d x} + 5 \frac{d y}{d x} - 2 y \frac{d y}{d x} - y + 2) (2 - x)

簡素化

(- 1) (y - 2) (3 - x - y) + (- x \frac{d y}{d x} + 5 \frac{d y}{d x} - 2 y \frac{d y}{d x} - y + 2) (2 - x) : x^{2} \frac{d y}{d x} + 2 x y + 2 x y \frac{d y}{d x} - 4 x - 7 x \frac{d y}{d x} + y^{2} + 10 \frac{d y}{d x} - 7 y - 4 y \frac{d y}{d x} + 10

= x^{2} \frac{d y}{d x} + 2 x y + 2 x y \frac{d y}{d x} - 4 x - 7 x \frac{d y}{d x} + y^{2} + 10 \frac{d y}{d x} - 7 y - 4 y \frac{d y}{d x} + 10