Question: partial derivatives of multivariate interpolation

In a related question to this answer ( How can we take partial derivatives of the multivariable fitting function (in the example above, the function B(a,b)?
It seems like the package CurveFitting only allows to numerically evaluate the fitted function when we use ArrayInterpolation, but I assume deep in the code there must be an analytic (I assume piecewise) fit since we use a spline fit for example. However, when I try to take partial derivates I get the following error
(Error, (in CurveFitting:-ArrayInterpolation) invalid input: xvalues are not specified correctly) 

Is there a way around that? if not, any suggestions on how I can obtain a multivariate interpolation that I can differentiate, using CurveFitting or any other package?

Please Wait...