It depends what you mean by understand. There are whole books written on logistic regression (probably most books) which never mention calculus and rarely mention matrix algebra (which is more pertinant IMHO). People run it everyday with no math background. So no you don't have to know these to understand the basis behind it or analyze with it. You might well have to have it in a statistics class (at least one in a college of math or statistics not many statistic programs that are applied) but you will use it very rarely outside the class.
When I have seen calculus mentioned in logistics regression it usually had to do with the maximum liklihood estimator that logistic regression uses. Not many pay a lot of attention to ML (unless of course it fails to coverge). Matrix algebra is generally a lot more important in treatment of nearly all statistics than calculus.