Given x = tan[x+y]
tan[x+y] = [tanx + tany] / [1 - tanxtany]
therefore
x = [tanx + tany]/[1- tanxtany]
applying ln
lnx = ln[tanx + tany] - ln[1- tanxtany]
1/x = [sec^2x + sec^2yf'(x)]/[tanx + tany] - [-sec^2xtany - sec^2ytanxf'(x)]/[1- tanxtany]
[1- tanxtany][tanx + tany] = x[1- tanxtany][sec^2x + sec^2yf'(x)] + x[tanx + tany][sec^2xtany + sec^2ytanxf'(x)]
[1- tanxtany][tanx + tany] = sec^2y.f'(x)[x - tanxtany + xtan^2x + xtanxtany] +sec^2x[x -tanxtany + xtanxtany + xtan^2y]
sec^2y.f'(x)[x - tanxtany + xtan^2x + xtanxtany] = [1- tanxtany][tanx + tany] - sec^2x[x -tanxtany + xtanxtany + xtan^2y]
f'(x) ={ [1- tanxtany][tanx + tany] - sec^2x[x -tanxtany + xtanxtany + xtan^2y] } / {sec^2y[x - tanxtany + xtan^2x + xtanxtany]}