x[y_,z_,k_]:=z+4ArcTan[y z]-k;

TangentPlane[x0_,y0_,z0_]:=Manipulate[
wx[x_,y_,z_]:=-1;
wy[x_,y_,z_]:=((4z)/(1+(((y z))^2)));
wz[x_,y_,z_]:=1+((4y)/(1+(((y z))^2)));
L[x_,y_,a_,b_,c_]:=c+((-1)/(wz[a,b,c]))(wx[a,b,c](x-a)+wy[a,b,c](y-b));
Show[
{Graphics3D[{
PointSize[Large],Point[{x0,y0,z0}]}],
ParametricPlot3D[{x[y,z,k],y,z},{y,-2,2},{z,-2,2}],
ParametricPlot3D[{u,v,L[u,v,x0,y0,z0]},{u,x0-.5,x0+.5},{v,y0-.5,y0+.5}],
ParametricPlot3D[{x0+t wx[x0,y0,z0],y0+t wy[x0,y0,z0],z0+t wz[x0,y0,z0]},{t,0,1}]
}
,AxesLabel->{x,y,z},Ticks->None,PlotRange->{{-2,5},{-2,2},{-2,3}}
]

,{{k,0},-5,5,1}];
TangentPlane[1+π,1,1]