Question

1.设 z=f(x^2-2y,xy), 其中f具有二阶连续偏导数,求 (a^2c)/(ax)0？

Answer 1

z = f(x^2-2y, xy)
∂z/∂x = 2xf'1+yf'2, ∂z/∂y = -2f'1+xf'2
∂²z/∂x² = 2f'1 + 2x(2xf''11+yf''12)+y(2xf''21+yf''22)
= 2f'1 + 4x^2f''11 + 4xyf''12 + y^2f''22
∂²z/∂x∂y = 2x(-2f''11+xf''12) + f'2 + y(-2f''21+xf''22)
= f'2 - 4xf''11 + (2x^2-2y)f''12 + xyf''22
∂²z/∂y² = -2(-2f''11+xf''12) + x(-2f''21+xf''22)
= 4f''11 - 4xf''12 + x^2f''22)