Derivatives of the associated function

I gave this up in graduate school because I did not know how to program. I think I could do it today but cannot remember the details of the first lemmas:

Let theta be angle between tangent at t and the chord from t to t+s on a [arc length parameterized] convex curve. Let r be the distance between points at t and t+s. Let psi be the angle between the chord and the tangent at t+s.

To find nth derivative in r of cos(theta) as r->0.

Let D = d()/dr, Dn the n-th such derivative and let L = lim as r->0. The problem is to find L*Dn( cos(theta) )

Lemma 1:

D(theta) = -tan(psi)/r

D(psi) = D(theta) - kappa*sec(psi)

where kappa is the curvature at t+s

Dn+1(kappa) = -Dn(kappa)*sec(psi)

Lemma 2:

L(theta) = 0
L(psi) = PI
L(Dn(kappa)) = ki  (a definition)

One needs to apply l'Hospitals rule and do a bit of algebra to solve for each Dn(theta). I find
L(D1(theta)) = -k0/2

L(D2(theta)) = k1/3 [or is it -(k0/2)^2??]

L(D3(theta)) = k0*k1/2

I propose that the successive answers are formulas in the ki's and you need a computer to find them, lacking cleverness.

Speculation: the successive answers, given in terms of polynomials in the ki's, form a Tauberian collection of polynomials, sufficient to reconstruct the complete original kappa - and hence the curve [usually!] If I had been a smart mathematician, I would have proved "Tauberian" without the explicit calculations.