Partial Derivative fy(x,y): What Is It?

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SUMMARY

The partial derivative of the function f(x,y) = x|y| with respect to y, denoted as fy(x,y), is determined by analyzing two cases based on the value of y. For y > 0, fy(x,y) = x, and for y < 0, fy(x,y) = -x. It is crucial to note that the function g(y) = |y| is not differentiable at y = 0, which results in f(x,y) having no partial derivative with respect to y along the line y = 0.

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squenshl
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Given that f(x,y) = x|y|
I know that fx(x,y) = |y| but what is fy(x,y).
Thanks heaps.
 
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Hint: Break it up into two cases: If y > 0, then f(x,y) = xy. If y < 0, then f(x,y) = -xy.
 
Note that g(y)= |y| is not differentiable at y= 0 so f(x,y)= x|y| has no partial derivative, with respect to y, on the line y= 0.
 

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