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Partial Derivative – Calculating second order partial derivatives to a function inside a square root in a ln function – Exercise 4323

Exercise 

Find  second order partial derivatives of the function

z(x,y)=\ln\sqrt{x^2+y^2}

Final Answer

z''_{xx} (x,y)=\frac{y^2-x^2}{{(x^2+y^2)}^2}

z''_{yy} (x,y)=\frac{x^2-y^2}{{(x^2+y^2)}^2}

z''_{xy} (x,y)=z''_{yx}(x,y)=\frac{-2xy}{{(x^2+y^2)}^2}

Solution

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