Should I Use Total or Partial Differentiation in Absolute Error Calculation?

[Chemist@]

The absolute error of the quantity p should be calculated.
p=a*b where a and b are two variables.
By taking ln on both sides:
lnp=lna+lnb, then differentiating:
dp/p=da/a+db/b
dp=abolute error
Should I have used total or partial differential in the differentiation step?

 

[ShayanJ]

Doesn't matter. da or \partial a or \Delta a, they all represent the error in measuring a.

 

[Chemist@]

Which is the most correct to use when I differentiate a function of two variables? Partial derivative?

 

[ShayanJ]

Well, now that I think, d is the correct one. Because you're actually differentiating \ln p w.r.t. p regardless of what p depends on! The same for other two logarithms.
Anyway, I've heard several methods for calculating absolute errors. But the one I prefer, is the following:
E[f(x_1,x_2,...,x_n)]=\sqrt{\sum_{k=1}^n (\frac{\partial f}{\partial x_k} dx_k)^2}
Because it obviously gives the maximum change in f for given changes in its arguments.