Does the difficulty of defining exactly what infinitesimals and differentials are denote a lack of rigor in standard real analysis?
For example, in an introductory course one may solve the differential equation such as $\frac{dy}{dx}=y$ by separation of variables by multiplying by $dx$ to obtain $\frac{dy}y=dx$ and integrating both sides, as if $dx$ and $dy$ were separate entities (infinitesimals) that can be manipulated algebraically (even though in the previous semester the students were typically taught that $\frac{dy}{dx}$ is an indivisible symbol rather than a fraction).
Do we really need to define what a differential is to operate with it or can we understand it as an important operator in the study of calculus without it representing a gap in the theory? I've already read that non-standard analysis came to cover these gaps, but is it really necessary in this context? When I say necessary I mean necessary to establish the theory with a strong, well-established rigor and without any informality.
This is a reasonable question though the choice of wording is not the best. The shortcoming of traditional analysis as it was formalized 150 years ago is not a failure at the level of rigor (on the contrary, it was an improvement on that score as noted in the comments) but rather a failure to incorporate infinitesimals, which up to that point had been the bread and butter of the practice of analysis for several centuries.
In the 1960s, modern theories of infinitesimals were developed that meet modern standards of precision, but the residual effect of the notion that "infinitesimals are unrigorous" still lingers.
Several editors made some useful comments under the question. My response is a bit too long to be included as a comment so I will include it here. Xander mentioned the issue of the a comparison of epsilon-delta and infinitesimal approaches. I would like to respond. Xander wrote:
While I agree with Xander's description of the construction of the hyperreals as "abstract", I would also point out that in the basic infinitesimal calculus sequence, there is no more need to present a construction of the hyperreal line than a construction of the real line in the non-infinitesimal approaches (which is in practice not usually constructed at this level). Keisler's textbook does a fine job explaining how to use the hyperreals in arguments; the construction itself can wait for a more advanced course. I should also clarify that the idea is not to replace epsilon-delta definitions entirely by ones using infinitesimals, but rather to explain the key notions of the calculus such as derivative, continuity, and integral in terms of infinitesimals, as a first step. Once the students understand these notions, they will have an easier time understanding the epsilon-delta paraphrases of these definitions.