TITLE: Newton vs Leibniz notation
QUESTION [16 upvotes]: I have often come across the cursory remarks made here and there in calculus lectures , math documentaries or in calculus textbooks that Leibniz's notation for calculus is better off than that of Newton's and is thus more widely used.
Though I have always followed Leibniz's notation( matter of familiarity, as that's what I have been taught) , but of late I had the idea of following Newton's notation just to see where I could get stuck just because of "notational" issues. 
Is there any limitation of Newton's notation that I might encounter while doing calculus ; and which may make it seem a bad idea to do calculus in Newton's notation?
Here "Leibniz notation" is $\frac{dy}{dx}$  for the derivative of $y$, and "Newton's notation" is $\dot{y}$ for the derivative of $y$.

REPLY [5 votes]: Wikipedia has a dedicated page on notations for differentiation, in short:

Leibniz $\frac{dx}{dt}$
Newton $\dot{x}$
Lagrange $x'(t)$

Leibniz's notation is suggestive, thanks to the cancelling of the differentials
in the chain rule:
$$
\frac{dy}{dt}=\frac{dy}{dx}\frac{dx}{dt}
$$
however great care must be taken, as this notation can also be misleading for higher order derivatives:
$$
\frac{d^2y}{dt^2}=\frac{d^2y}{dx^2}\frac{dx^2}{dt^2}=\frac{d^2y}{dx^2}\left(\frac{dx}{dt}\right)^2
$$
which is wrong, the right formula is:
$$
\frac{d^2y}{dt^2}=\frac{d^2y}{dx^2}\left(\frac{dx}{dt}\right)^2+\frac{dy}{dx}\frac{d^2x}{dt^2}
$$
You have not this problem with Lagrange's notation:
$$
y(x(t))''=(y'(x(t))x'(t))'=y''(x(t))(x'(t))^2+y'(x(t))x''(t)
$$
These notation problems are well known when teaching differential calculus, see:
H. Poincaré, La Notation Différentielle et l'enseignement (pdf)
J. Hadamard, La notion de différentielle dans l'enseignement (pdf)
unfortunately both in French, however you can find an English translation of Hadamard's article here.
You can also see:
Differentials, higher-order differentials and the derivative in the Leibnizian calculus (pdf)