Taking the limit of a derived function

I need to evaluate an expression of the form

$$\int_0 ^ a dx \left[\frac{\partial}{\partial \alpha} \left[ \frac{\partial^n}{\partial \beta^n} \left[\frac{\partial}{\partial \gamma} g(x,\alpha,\beta,\gamma) \right]_{\gamma \to 0} \right]_{\beta \to 0} \right]_{\alpha \to 0},$$

with g given by

$$g(x,\alpha,\beta,\gamma) = \det \left( \alpha A(x) + \beta B(x) + \gamma C(x) + D(x) \right)$$

where A, B, C and D are general (known) matrices. Ultimately, the evaluation will be performed numerically, but since performance is a major concern, I am curious to find a "smart way". I am not very much into math, but so far I have though about two different approaches:

1) For each value of x, evaluate g for different values of $\alpha, \beta$ and $\gamma$. The differentiation can then be performed trivially numerically, but i am not really sure about how to do the limits.

2) Approximate the determinant by expansion in Chebyshev (or some other kind of) polynomials. In this case, the differential and limit operations will be easier to do, but i am not about the accuracy of this approach.