As @BenCrowell mentioned, the transfer principle proves that direct algebraic manipulation of infinitesimals in single variable calculus is allowed.

But I stumbled upon [this post in theshapeofmath.com](http://www.theshapeofmath.com/oxford/physics/year1/calc/sepdiff), which shows how things can break when switching to multiple variables. It provides the following basic example of a possible error when handling partial derivatives (the same example also appears as [an answer](https://mathoverflow.net/a/73503/49215) in the MO question linked to by @AmirAsghari), where given an implicitly defined function
$F(x,y)=0$ , we can algebraically get an expression with the wrong sign:
$$\frac{dy}{dx}=\frac{\partial F}{\partial x}\frac{\partial y}{\partial  F}=\frac{\frac{\partial F}{\partial x}}{\frac{\partial F}{\partial y}}.$$

While the correct expression is:
$$\frac{dy}{dx}=-\frac{\frac{\partial F}{\partial x}}{\frac{\partial F}{\partial y}}.$$