TITLE: Getting the original function from a derivative
QUESTION [0 upvotes]: i know that $f_x = 4x^3 + 4xy $ is the partial derivative of the function $ f = x^4 + 2x^2y $
But how is the original function  $ f = x^4 + 2x^2y $ found from the partial derivative ?
As in, how do i reverse the process?

REPLY [5 votes]: Rule:  We see $y$ as a constant. Hence we have
$$f(x)=\int f_x dx=x^4+2x^2y+g(y)$$
where $g(y)$ is any function on $y$.
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