Hey Okkes!

Thanks for the response. My only concern was, how will I know which roots correspond to which derivatives? This is where I thought using the "AppendTo" command would be useful. This is certainly a step in the right direction and exactly the output I was trying to produce. So thanks again!

Brandon

I got it! This is what I was going for:

Clear[x, roots, x, n, d];
f[x_] = x^8 - 3 x^5 + x - 1;
n = 5;
der = D[f[x], {x, #}] & /@ Range[n];
roots = Table[NSolve[der[[#]] == 0, x, Reals] & /@ Range[n]];
Grid[{der, roots} // Transpose]

output:

{
 {1 - 15 x^4 + 8 x^7, {{x -> -0.5}, {x -> 0.518012}, {x -> 1.22064}}},
 {-60 x^3 + 56 x^6, {{x -> 0}, {x -> 0}, {x -> 0}, {x -> 1.02326}}},
 {-180 x^2 + 336 x^5, {{x -> 0}, {x -> 0}, {x -> 0.812165}}},
 {-360 x + 1680 x^4, {{x -> 0}, {x -> 0.598408}}},
 {-360 + 6720 x^3, {{x -> 0.376974}}}
}

Thanks again for your help because I was able to combine what you gave me along with some ideas I was messing around with. Great teamwork and much appreciated!

Brandon