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Partial Derivative – Calculating second order partial derivatives to a function inside a square root – Exercise 4320

Exercise 

Find second order partial derivatives of the function

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

Final Answer

z''_{xx} (x,y)=\frac{y^2}{{(x^2+y^2)}^{\frac{3}{2}}}

z''_{yy} (x,y)=\frac{x^2}{{(x^2+y^2)}^{\frac{3}{2}}}

z''_{xy} (x,y)=z''_{yx}(x,y)=\frac{-xy}{{(x^2+y^2)}^{\frac{3}{2}}}

Solution

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