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**AllanCuz** $\displaystyle grad f(x,y) = \nabla f(x,y) = \frac{ \partial f }{ \partial x } \hat i + \frac{ \partial f }{ \partial y } \hat j $

We are given $\displaystyle \frac{ \partial f }{ \partial x } = y^2e^4x+2xe^3y $ so we need to find a Potential Function such that $\displaystyle \frac{ \partial f }{ \partial y } = G(x,y) $

To do this we will find F(x,y) by integration of the first part. Then find a constant such that our criteria is satisfied

$\displaystyle \frac{ \partial f }{ \partial x } = y^2e^4x+2xe^3y $

$\displaystyle \partial f = y^2e^4x+2xe^3y \partial x $

$\displaystyle f(x,y) = \int y^2e^4x+2xe^3y dx = \frac{ y^2e^4x^2 }{2} + x^2e^3y + Q(y) $ where Q(y) is a function only dependant on y.

Therefore,

$\displaystyle \frac{ \partial f }{ \partial y } = G(x,y) = ye^4x^2 + x^2e^3 + Q`(y) $

To get to that step we simply differentiate our potential function with respect to y!