求下列函数的二阶偏导数 1）z=x^4+3*x^2*y+y^3 2）z=xln(x+y)

题目

求下列函数的二阶偏导数 1）z=x^4+3*x^2*y+y^3 2）z=xln(x+y)
求下列函数的二阶偏导数
1）z=x^4+3*x^2*y+y^3
2）z=xln(x+y)

答案

z=x^4+3x²y+y³
∂z/∂x = 4x³+6xy
∂z/∂y = 3x²+3y²
∂²z/∂x² = 12x²+6y
∂²z/∂x∂y = 6x
∂²z/∂y² = 6y
----------------------------
z=xln(x+y)
∂z/∂x = ln(x+y) + x/(x+y)
∂z/∂y = x/(x+y)
∂²z/∂x² = 1/(x+y) + [(x+y)-x]/(x+y)² = 1/(x+y) + y/(x+y)² = (x+2y)/(x+y)²
∂²z/∂x∂y = 1/(x+y) - x/(x+y)² = y/(x+y)²
∂²z/∂y² = - x/(x+y)²