c = Plot[y*Sin[8] + 8*Sin[y] + 8 + y == 0, {y, -2 Pi, 2 Pi}];
cc = ContourPlot[y*Sin[x] + x*Sin[y] + x + y == 0, {x, -20, 20}, {y, -20, 20}];
Show[c, cc]

implicit = D[y[x]*x*Sin[x] + x*Sin[y[x]] + x + y[x] == 0, x]

nsol = NSolve[y*Sin[8] + 8*Sin[y] + 8 + y == 0 && -20 < y < 20, y]

(*    {{y -> -5.8196}, {y -> -2.84462}, {y -> -0.891837}}    *)

pts = Thread[List[8, y /. nsol]]

(*    {{8, -5.8196}, {8, -2.84462}, {8, -0.891837}}    *)

solys = Solve[D[y[x]*Sin[x] + x*Sin[y[x]] + x + y[x] == 0, x], y'[x]]

(*   {{Derivative[1][y][x] -> (-1 - Sin[y[x]] - Cos[x] y[x])/(
       1 + x Cos[y[x]] + Sin[x])}}   *)

yt[x_] = y'[x] /. First@solys

grad = (yt[8] /. y[8] -> #[[2]]) & /@ pts

(*   {-0.250838, 0.198087, -0.0501247}   *)

ContourPlot[{y*Sin[x] + x*Sin[y] + x + y == 0,
  Thread[y == Plus[y /. nsol, grad (x - 8)]]} // Flatten //
  Evaluate, {x, -20, 20}, {y, -20, 20}, MaxRecursion -> 4,
  Epilog -> Point@pts]