Question about how to run first difference in Panel data ?

My baseline equation
y = alpha (x) + year fixed effect + e (1)
e_hat = beta (z) + country fixed effect (2)
I saved the residuals from eq (1) and then I use e_hat to regress with a particular variable (z) in eq (2) .

My question: how can I run first difference for eq (2) ?
Thank you !

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