I've found a slightly easier implementation of the derivative for you based on this link. It essentially says that
In MATLAB, this would look like
function dJndx = dbesselj(n,x)
% DBESSELJ A function that will generically calculate the
% the derivative of a Bessel function of the first
% kind of order n for all values of x.
%
% Example usage: dJndx = dbesselj(n,x);
%
% INPUT ARGUMENTS
% ================
% n Order of the Bessel function of the first kind
% x Input variable to Bessel function
%
% OUTPUT ARGUMENTS
% ================
% dJndx Derivative of nth order Bessel function of the first
% kind at all values of x
dJndx = n*besselj(n,x)./x - besselj(n+1,x);
end
Hopefully this answers your question, let me know if this helps!
