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Multi-variable first-order perturbation analysis?

I have a half dozen functions f1, f2, ?, each of which depends on some 
or all of half a dozen variables  x1, x2, ?, all of these functions 
pretty vanilla in character.  

Objective is to obtain the same number of first-order perturbation 
expansions  df1, df2, ?, where  df1 means all the relevant derivatives

   (df1/dx1) * dx1  +  (df2/dx2) * dx2 + . . . 

evaluated at initial values  x1=x10, x2=x20, ? and so on -- all of this 
totally symbolic in character, and with *no* cross-products dx1 dx2 or 

I know I can code this various ways -- but what's the "cleanest" way to 
accomplish this?

[Notes:  The results for df1, df2, ? don't have to be neatly readable, 
e.g., the terms multiplying  dx1  for a given  dfi  don't all have to be 
neatly collected inside a single set of brackets; the resulting 
expressions just have to be correct.  And, by "cleanest" I don't 
necessarily mean the tersest, most arcane way of coding this.]

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