TITLE: Strange Method of Differentiating $x^2$
QUESTION [7 upvotes]: A book by the name of Calculus Super Textbook teaches the following method for differentiating $x^2$ with respect to $x$: Take the equation: $$y=x^2$$ The author reasons that increasing both $x$ and $y$ by a differential amount preserves equality, so: $$y+dy=(x+dx)^2$$
Expanding the right-hand side gives: $$y+dy=x^2+2xdx+dx^2$$ Subtract off the original $y=x^2$: $$dy=2xdx+dx^2$$ The author reasons that an infinitesimal change in $x$ multiplied by an infinitesimal change in $x$ is so small to be effectively zero, so any powers of $dx$ higher than $1$ are dropped: $$dy=2xdx$$ Then, dividing both sides by $dx$: $${dy \over dx}=2x$$ This is nothing like the methods I've been taught for finding a derivative (which I understand to be the limit of the difference quotient), but it at least appears to work for polynomial functions. Is this a valid method that I'm not aware of? If so, how would you use this to differentiate more complex functions like sine and cosine?

REPLY [7 votes]: This is calculus based on differentials, which is different from the limits-based calculus that is prevalent these days.  However, despite what some commenters (and contemporaries of Newton and Leibniz) feel, this is perfectly sound logically.  If you'd like to see some details of how to work with infinitesimals, I recommend Keislers' Calculus, which he has kindly placed online for free.