Question: Minimum positive real root(s) of large matrix determinant

Please explain a method (including corresponding commands) to find first positive real root(s) of a large matrix determinant, briefly. Since each entry or element of the matrix involves with the large formula, probably Newton Iterative method is suitable. The main question is, how it is possible to define some matrices like the Hessian or gradient matrices, when calculating parametric solution of determinant is a very time consuming procedure.

Let assume that only one independent variable exists. After evaluating matrix value at a certain amount of independent variable, the determinant can be calculated swiftly. The problem is that the derivative of determinant in each step will be calculated slowly. How one can rectify this deficiency in Newton Iterative method? For example, is there another iterative method to find derivative of determinant at a certain amount of independent variable to use it in the Newton iterative method?