TITLE: Is $d^2y/dx$ a valid mathematical notation?
QUESTION [8 upvotes]: I have often seen "the second derivative of y with respect to x" written as $${d^2y\over dx^2},$$ but I don't understand the reason for this notation. I have always seen it written as $${d^2y\over dx^2},$$ and never as $${d^2y\over dx}.$$ Would $d^2y\over dx$ be a valid notation as well, and would it have a different meaning from $d^2y\over dx^2$?

REPLY [2 votes]: As in the other answers, the main response to the question is that probably something is wrong... since the basic Leibniz notation would never write ${d^2y \over dx}$.
The only possible sense would be an "infinitesimal" change in the derivative ${dy\over dx}$, in Leibniz' notation indeed $d{dy \over dx}$, though I've seen this only very rarely, and it would rarely be useful.
Nevertheless, there are some manipulations to solve differential equations that do exactly make use of similar ideas. For example, in one dimension, Newton'e inverse square law ${d^2y\over dx^2}={-1\over y^2}$ can be rewritten (usefully!) as
$$
{-1\over y^2} = {d^2 y\over dx^2} = {d\over dx}{dy\over dx}
= {d(y')\over dx} = {dy'\over dy}{dy\over dx} = {dy'\over dy}\,y'
$$
Then ${-dy\over y^2}=y'\,dy'$ gives ${1\over y}={1\over 2}(y')^2$, and so on. :)