Mark M. answered 10/18/19
Retired math prof. Calc 1, 2 and AP Calculus tutoring experience.
xy2 + ln(xy) = e4 + 3
xy2 + lnx + lny = e4 + 3
y2 + 2xy(dy/dx) + 1/x + (1/y)(dy/dx) = 0
dy/dx(2xy + 1/y) = -y2 - 1/x
So, at the point (e2 , e), we have dy/dx(2e3 + 1/e) = -e2 - 1/e2
dy/dx = (-e2 - 1/e2) / (2e3 + 1/e) = (-e4 - 1) / (2e5 + e)