I am trying to find the maximum likelihood estimate of the parameters for the t-copula. Ideally I'd want to use a gradient-based method for optimization. However, I am having some difficulty in finding the partial derivative of the inverse of the standard, univariate t-CDF with v degrees of freedom with respect to v.

Though this question seems straightforward to me, I have yet to find a resource that provides a solution. From what I have seen, the t-copula literature tends to overlook any concerns about estimating v, instead focusing on estimating the shape matrix of the joint distribution.

Are there any suggestions or resources for ways that might I approach this partial derivative?