ePartialDifferentiation

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

$\displaystyle f(x,y) = x^{2}e^{-y}$

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 $y$ .

$\displaystyle f_{x}(x,y)$	$\displaystyle =$	$\displaystyle (\frac{\partial x^2}{\partial x})e^{-y} = 2xe^{-y}$
$\displaystyle f_{y}(x,y)$	$\displaystyle =$	$\displaystyle x^2(\frac{\partial e^{-y}}{\partial y}) = -x^2 e^{-y}$

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