The 3D visualization has an arrowhead size problem.

Show[Plot3D[{f[x, y]}, {x, -5, 5}, {y, -8, 8}, PlotRange -> {-5, 5}, 
  AspectRatio -> 1, AxesLabel -> Automatic],
 Graphics3D[{PointSize[Large], Black, Point[{P, P + Normalize[Q]}]}],
 Graphics3D[{Arrowheads[0.02], Arrow[{P, P + Normalize[Q]}]}], 
 ImageSize -> Large]

I am not sure about the direction -- I did not check that. Are you trying to plot a normal to the surface?

Regards

Neil

Mitchell,

Your problem is that you are computing the slope at point {-2,3} as Q2 -- this is correct, however, when you plot Q2 you plot it as a point. It is a direction -- your plot must add P2 plus some multiple of Q2. The multiple determines the vector length. For example, use P2+Normalize[Q2].

Using your example:

Show[ContourPlot[f[x, y], {x, -5, 10}, {y, -5, 10}],
 Graphics[{{PointSize[Large], Black, 
    Point[{P2, P2 + Normalize[Q2]}]}}],
 Graphics[Arrow[{P2, P2 + Normalize[Q2]}]],
 VectorPlot[Evaluate[Grad[f[x, y], {x, y}]], {x, -5, 10}, {y, -5, 10},
   VectorPoints -> 8, AxesLabel -> Automatic]]