First derivative of (normal) associated Legendre polynomials

Computing accurate derivatives of the associated Legendre polynomials can be tricky. Even in advanced texts, they are usually written as recurrence relations and/or with (normalization) factors involving factorials.
A naive software implementation will therefore quickly run into the limits of IEEE754 double-precision, resulting in NaN/inf or significant loss of precision, already at very low degree N.
LEGENDRE_DERIVATIVE is a fully vectorized, numerically stable and robustly validated implementation of the derivative computation. It allows fast and accurate computations of the derivatives for any degree N.

It works the same as MATLAB's own LEGENDRE, except it does not compute the polynomial values, but the values of the derivatives.