(d^2)/(d{x)d{y}}(({z}(x,y))(({x}^4{y}^2+{x}+{y})({x}-{y})))

解

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

z (x, y) (- 12 y^{2} x^{3} + 4 y^{2} x^{3} \frac{d}{d y} (x) + 8 y x^{4} + \frac{d}{d y} (x) - 1 + 2 y x^{4} + \frac{d}{d y} (x) + 1)

解答ステップ

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

\frac{d}{d x} (z (x, y) (x^{4} y^{2} + x + y) (x - y)) = z (x, y) ((4 x^{3} y^{2} + 1 + \frac{d}{d x} (y)) (x - y) + (1 - \frac{d}{d x} (y)) (x^{4} y^{2} + x + y))

= \frac{d}{d y} (z (x, y) ((4 x^{3} y^{2} + 1 + \frac{d}{d x} (y)) (x - y) + (1 - \frac{d}{d x} (y)) (x^{4} y^{2} + x + y)))

\frac{d}{d y} (z (x, y) ((4 x^{3} y^{2} + 1 + \frac{d}{d x} (y)) (x - y) + (1 - \frac{d}{d x} (y)) (x^{4} y^{2} + x + y))) = z (x, y) (- 12 y^{2} x^{3} + 4 y^{2} x^{3} \frac{d}{d y} (x) + 8 y x^{4} + \frac{d}{d y} (x) - 1 + 2 y x^{4} + \frac{d}{d y} (x) + 1)

= z (x, y) (- 12 y^{2} x^{3} + 4 y^{2} x^{3} \frac{d}{d y} (x) + 8 y x^{4} + \frac{d}{d y} (x) - 1 + 2 y x^{4} + \frac{d}{d y} (x) + 1)