z=x*ln(x+y)对x,y,xy求二阶偏导数
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∂z/∂x=ln(x+y) +x/(x+y),
∂z/∂y= x/(x+y),
所以
∂²z/∂x²
=∂[ln(x+y) +x/(x+y)] / ∂x
=1/(x+y) + [(x+y) -x]/(x+y)²
∂²z/∂x∂y
=∂[ln(x+y) +x/(x+y)] / ∂y
=1/(x+y) - x /(x+y)²
∂²z/∂y²
=∂ [x/(x+y)] /∂y
= -x /(x+y)²
∂z/∂y= x/(x+y),
所以
∂²z/∂x²
=∂[ln(x+y) +x/(x+y)] / ∂x
=1/(x+y) + [(x+y) -x]/(x+y)²
∂²z/∂x∂y
=∂[ln(x+y) +x/(x+y)] / ∂y
=1/(x+y) - x /(x+y)²
∂²z/∂y²
=∂ [x/(x+y)] /∂y
= -x /(x+y)²
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