From Apostol - Introduction to analytic number theory (Theorem 3.3) we have $$ x\geq1, \sum_{n\leq x}d(n)=x\log x+(2\gamma-1)x+O(\sqrt{x}):=E(x), $$ I want to differentiate $E$ -- to get a rough estimate for $d(n)$ -- but I don't know how to deal with the big-o part (even by proceeding with its rigorous definition doesn't get me anywhere).
- How to differentiate a function with a big-o?
Thanks
As commenters said, a bound on a function does not give any bound in its derivative.
However, here we have partial sums of a nonnegative function of an integer argument. So at least something can be said (though it will be trivial):
$$\sum_{n\leq x}d(n)=x\log x+(2\gamma-1)x+O(\sqrt{x}) \tag{1}$$ implies $$d(x) \le x\log x+(2\gamma-1)x+O(\sqrt{x}) \tag2$$ Obviously, this is totally useless since by definition of $d$ we have $d(x) \le x$ anyway.
But if we didn't know anything about $d$ except (1) and $d\ge 0$, then (2) is the best bound one can have, since it is conceivable that $d$ is zero for $1,\dots,x-1$ and all of the partial sum in (1) is just $d(x)$.