z=x^4+3x²y+y³
∂z/∂x = 4x³+6xy
∂z/∂y = 3x²+3y²
∂²z/∂x² = 12x²+6y
∂²z/∂x∂y = 6x
∂²z/∂y² = 6y
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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