Hello,
What do you mean by "Can Mathematica graph a double derivative?"
Can you do that? If so, graph y''.....
Ah, you can't? Exactly, you have to know y to calculate y'' and then you can plot it.
So I think you should get a solution for your c and then you may be able to plot almost everything.
But be careful what you are requesting Mathematica to do. Here e.g. I don't know the reason for the error message, I could fix these things in earlier examples but here it doesn't work, so the solution may be totally wrong.
e1 = D[c[x, t], t] == fD D[c[x, t], x, x] + k c[x, t] (c0 - c[x, t])
e2 = c[x, 0] == 0
e3 = D[c[x, t], x] == .3 /. x -> 0
e4 = c[0, t] == c0 (1 - Exp[-15 t])
vals = {fD -> .03, k -> .9, c0 -> 1.2}
sol = NDSolve[({e1, e2, e3, e4} /. vals), c, {t, 0, 5}, {x, 0, 10}][[1, 1]]
f = c /. sol
Plot3D[f[x, t], {x, 0, 10}, {t, 0, 5}, PlotRange -> All]
Plot3D[Derivative[0, 1][f][x, t], {x, 0, 10}, {t, 0, 5},PlotRange -> All]
Plot3D[Derivative[2, 0][f][x, t], {x, 0, 10}, {t, 0, 5}, PlotRange -> All]