Convert BSplineFunction into two Interpolating Functions

When manipulating B-splines in this manner, it is often convenient to fall back on the definitions. Luckily, since Mathematica supplies the function BSplineBasis[], using the definitions are easy:

pts = {{0, 0}, {1, 1}, {2, -1}, {3, 0}, {4, -2}, {5, 1}};

n = 3; (* B-spline degree *)
m = Length[pts];
(* clamped uniform knots for B-spline *)
knots = {ConstantArray[0, n + 1], Range[m - n - 1]/(m - n), ConstantArray[1, n + 1]}
        // Flatten;

{xu, yu} = Transpose[pts];
bs = BSplineFunction[pts, SplineDegree -> n];

(* B-spline component functions *)
f[t_] = xu.Table[BSplineBasis[{n, knots}, i - 1, t], {i, Length[pts]}];
g[t_] = yu.Table[BSplineBasis[{n, knots}, i - 1, t], {i, Length[pts]}];

Compare:

{ParametricPlot[bs[t], {t, 0, 1}, Axes -> None, Frame -> True,
                Epilog -> {Directive[AbsolutePointSize[5], Red], Point[pts]}], 
 ParametricPlot[{f[t], g[t]}, {t, 0, 1}, Axes -> None, Frame -> True,
                Epilog -> {Directive[AbsolutePointSize[5], Red], Point[pts]}]}

One can now plot the component functions as needed:

Plot[{f[t], g[t]}, {t, 0, 1}, Axes -> None, Frame -> True]

or use derivatives:

With[{t = 1/3}, g''[t]/Sqrt[f'[t]^2 + g'[t]^2]]
   48/Sqrt[41]

What about trying something like the following!

pts = {{1, 1}, {2, 3}, {3, -1}, {4, 1}, {5, 0}};
f = BSplineFunction[pts];
x[t_?NumericQ] := Module[{val}, val = f[t]; First@val];
y[t_?NumericQ] := Module[{val}, val = f[t]; Last@val];

Check it!

Plot[{x[t], y[t]}, {t, 0, 1}, Frame -> True]

Now the value you are looking for.

nf[t_?NumericQ] := y''[t]/Sqrt[x'[t]^2 + y'[t]^2];
nf''[0.5]

-758.244

BR