How to interpolate this table I have a 3D function(al) f whose independent variables are A,C,D and E.
Various tables are provided to show the function values.
For each individual table, value of A is a constant.
The nth table is provided for  A + nk where k is a constant.
The structure of each table is as follows:
Column 1 = X(value of the first dependent variable)
Column 2 = Y(value of the second dependent variable)
Column 3 = Z(value of the third dependent variable)
Column 4 = dXC(change of value of the first dependent variable, for C+dC, where dC is the change in C, and D,E remains constant)
Column 5 = dXD(change of value of the first dependent variable only, for D+dD, where dD is the change in D, and C,E remains constant)
Column 6 = dXE(change of value of the first dependent variable only, for E+dE, where dE is the change in E, and C,D remains constant)
Column 7 = dYC(change of value of the second dependent variable only, for C+dC, where dC is the change in C, and D,E remains constant)
Column 8 = dYD(change of value of the second dependent variable only, for D+dD, where dD is the change in D, and C,E remains constant)
Column 9 = dYE(change of value of the second dependent variable only, for E+dE, where dE is the change in E, and C,D remains constant)
The changes in respective independent variables are same across tables.
Each row is given for uniform incremental values of X.
For each row Y and Z are calculated for same values of C,D,E for which X is calculated.
That is if X = f1(a1,c1,d1,e1), then Y = f2(a1,c1,d1,e1) and Z = f3(a1,c1,d1,e1).
Now my problem:
I want to calculate the values of X,Y and Z for intermediate values in C,D and E.
If I apply small changes in C,D and E simultaneously, what will be values of X, Y and Z.
(These changes are smaller than dC,dD and dE respectively).
How can I approach this problem.
What subject of mathematics deals with these kind of problems.

This appears to me to be an interpolation problem. In fact, this sounds like a challenging interpolation problem (I haven't done multivariate interpolation myself, and I imagine that some software can do it for you; but I am familiar with univariate interpolation, and it has enough complexity on its own in my opinion).
I suspect that you'll want to use "Catmull-Rom splines," well-studied interpolation points that work in any dimension. 
So to answer your question, this area of math is 'multivariable interpolation.'
