f[x_,y_,z_]:=Sin[x*y]*z-x^2*y;
Dx[f_][x_,y_,z_]:=D[f[u,v,w],u] /. {u->x,v->y,w->z}
Dy[f_][x_,y_,z_]:=D[f[u,v,w],v] /. {u->x,v->y,w->z}
Dz[f_][x_,y_,z_]:=D[f[u,v,w],w] /. {u->x,v->y,w->z}
grad[f_][x_,y_,z_]:={Dx[f][x,y,z],Dy[f][x,y,z],Dz[f][x,y,z]};
L[x_,y_,z_]:=f[1,1,1]+grad[f][1,1,1].{x-1,y-1,z-1}
