As @BenCrowell mentioned, the transfer principle proves that direct algebraic manipulation of infinitesimals in single variable calculus is allowed.
But I now stumbled upon this post in theshapeofmath.com on the subject shows how things 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 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: $$\frac{dy}{dx}=-\frac{\frac{\partial F}{\partial x}}{\frac{\partial F}{\partial y}}.$$
