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### 40.5 Evaluation of B-spline derivatives

Function: int gsl_bspline_deriv_eval (const double x, const size_t nderiv, gsl_matrix * dB, gsl_bspline_workspace * w)

This function evaluates all B-spline basis function derivatives of orders 0 through nderiv (inclusive) at the position x and stores them in the matrix dB. The (i,j)-th element of dB is d^jB_i(x)/dx^j. The matrix dB must be of size n = nbreak + k - 2 by nderiv + 1. The value n may also be obtained by calling `gsl_bspline_ncoeffs`. Note that function evaluations are included as the zeroth order derivatives in dB. Computing all the basis function derivatives at once is more efficient than computing them individually, due to the nature of the defining recurrence relation.

Function: int gsl_bspline_deriv_eval_nonzero (const double x, const size_t nderiv, gsl_matrix * dB, size_t * istart, size_t * iend, gsl_bspline_workspace * w)

This function evaluates all potentially nonzero B-spline basis function derivatives of orders 0 through nderiv (inclusive) at the position x and stores them in the matrix dB. The (i,j)-th element of dB is d^j/dx^j B_(istart+i)(x). The last row of dB contains d^j/dx^j B_(iend)(x). The matrix dB must be of size k by at least nderiv + 1. Note that function evaluations are included as the zeroth order derivatives in dB. By returning only the nonzero basis functions, this function allows quantities involving linear combinations of the B_i(x) and their derivatives to be computed without unnecessary terms.