I want to use the first-order and seconds-order partial derivatives of t copula in matlab, how to find their formulas? I want to use the first-order and seconds-order partial derivatives of t copula in matlab, however, I cannot use diff() function to get its first-order and seconds-order partial derivatives. Who knows where I can find their formulas or the matlab codes? Thanks very much!
 A: The bivariate t copula (with $\nu$ degrees of freedom and correlation $\rho$) is 
$$C_t(u,v,\rho)=\frac{\Gamma(\frac{\nu+2}{2})}{\Gamma(\frac{\nu}{2})\sqrt{\pi \nu}\sqrt{1-\rho^2}}\int_{-\infty}^{t_\nu^{-1}(u)}\int_{-\infty}^{t_\nu^{-1}(v)}\left[1+\nu^{-1}\left(x^2-2\rho xy+y^2\right)\right]^{-\frac{\nu+2}{2}}\,dx\,dy,$$
where $t_\nu^{-1}$ is the inverse of a standard univariate $t$ cummulative distribution function.
Find the partial derivatives with respect to $u$ and $v$ by differentiating with Leibniz's rule and using
$$\frac{d}{du}t_\nu^{-1}(u)= \left[t_\nu'(t_\nu^{-1}(u))\right]^{-1}\\=\frac{\Gamma(\frac{\nu+1}{2})}{\Gamma(\frac{\nu}{2})\sqrt{\pi\nu}}\left(1+\frac{t_\nu^{-1}(u)}{\nu}\right)^{-\frac{\nu+1}{2}}.$$
A: -In the denominator of the t Copula expression is pi times nu only, not the square root
-You only need the fundamental theorem of calculus not Leibnitz's rule
-In the last expression the exponent -1 is forgotten and not taken into account
If this is meant to be a source of knowledge, it needs to be more accurate
