find the partial derivatives fx and fy (a) f(x,y)= sqrt(x/y) + sqrt(y/x)thanks for any help...

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lfryerda | High School Teacher | (Level 2) Educator

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To find the partial derivative `f_x` , we pretend that y is a constant and to find the partial derivative `f_y` , we pretend that x is a constant.

This means that for the function `f(x,y)=x^{1/2}y^{-1/2}+x^{-1/2}y^{1/2}` ,

`f_x=1/2x^{-1/2}y^{-1/2}-1/2x^{-3/2}y^{1/2}`    take largest negative exponent from denominator

`=1/{2x^{3/2}y^{1/2}}(x-y)`

`={x-y}/{2x^{3/2}y^{1/2}}`

For the partial derivative `f_y` , notice that interchanging x and y gives the same as `f_x` through the symmetry of the variables.

This means that `f_y={y-x}/{2y^{3/2}x^{1/2}}`.

The partial derivatives are `f_x={x-y}/{2x^{3/2}y^{1/2}}` and `f_y={y-x}/{2y^{3/2}x^{1/2}}` .