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Example Find the partial derivatives $f_{x}, f_{y}$ of the following function.

$\displaystyle f(x,y) = \sqrt{x^2 + y^2}$

Answer The partial derivative with respect to $x$ written $f_{x}$. To find $f_{x}$, we treat all variables except $x$ as constants. Then differentiate with respect to $x$．Similarly, the partial derivative with respect to $y$ is given by $f_{y}$ and it can be found by treating all variables except $y$ as constants. Then differentiate with respect to $y$.


$\displaystyle f_{x}(x,y)$ $\displaystyle =$ $\displaystyle \frac{1}{2\sqrt{x^2 + y^2}}\frac{\partial (x^2+y^2)}{\partial x}$  
  $\displaystyle =$ $\displaystyle \frac{1}{2\sqrt{x^2+y^2}}(2x) = \frac{x}{\sqrt{x^2 + y^2}}$  
$\displaystyle f_{y}(x,y)$ $\displaystyle =$ $\displaystyle \frac{1}{2\sqrt{x^2 + y^2}}\frac{\partial (x^2+y^2)}{\partial y}$  
  $\displaystyle =$ $\displaystyle \frac{1}{2\sqrt{x^2+y^2}}(2y) = \frac{y}{\sqrt{x^2 + y^2}}$