Is $\dfrac{dx}{dt}$ a fraction or not? I am new to calculus and during my mathematics class my sir defined $\dfrac{dx}{dt}$ as $$dx/dt=\lim_{t\to t_1}\dfrac{f(t)-f(t_1)}{t-t_1}$$ and my sir made a clear statement that 

$\dfrac{dx}{dt}$ is not a fraction it only behaves like a fraction!

(it means $\dfrac{dx}{dt}$ is just a notation to represent that big limit!) and he made a statement that

$dx$ or $dt$ does not have any meaning it is just $\dfrac{d}{dt}(x)$ which has meaning but we treat it as $\dfrac{dx}{dt}$.

but at same time my physics sir, to derive velocity he stated 

let the particle be at position $x$ at time $t$ and after an infinitesimal change in position and time it reaches $x+dx$ at time $t+dt$. Now velocity is $\dfrac{displacement}{time}$ , so we will get $v=\dfrac{dx}{dt}$ 

and this expression purely tells that $\dfrac{dx}{dt}$ is a fraction!
Now i don't know who is correct, so please help! 
 A: What an interesting question! It depends. In modern calculus, $\frac{df}{dt}$ is just a symbol for the derivative
$$\lim_{h \to 0} \frac{f(t+h)-f(t)}{h}$$
As a matter of fact, mathematicians prefer different notations for the derivative of a function $f$, as $D f$ or $f'$. 
But the above definition of derivative became rigorous only when the concept of limit became rigorous, and this happened only with Weierstrass around 1870 and with his (in)famous "epislon-delta" notation. But we all know that calculus was already used (and maybe invented), in a primitive form, by Newton and Leibniz in the 17th century! 
Newton and Leibniz thought of derivatives in different ways: for Newton, it represented a fluxion, a rate of flux or change. For Leibniz, it was the ratio of infinitesimal (really, really small) differences, a differential quotient. In fact, it was Leibniz who first introduced the "quotient" notation $\frac{df}{dt}$! 
For example, Leibniz argued that $\frac{d(x^2)}{dx}=2x$ because
$$\frac{(x+dx)^2-x^2}{dx} = \frac{2 x dx + dx^2}{dx} = 2 x + dx$$
and, since $dx$ is infinitesimal, we can ignore it: Q.E.D. ... or not?
The point is that a reasoning such as this is not rigorous enough and can lead to every kind of inconsistencies ($dx$ cannot be exactly $0$, otherwise the quotient would not exist...). So, after a while, mathematicians said goodbye to those treacherous "infinitesimals" and adopted once and for all the more rigorous notation of Weierstrass.
In physics, we are way more laid-back kind of guys: sometimes mathematical rigor just isn't our thing (I remember one of my professors saying this once: "This theorem is false...but we are going to use it anyway!"). We secretly stuck to Leibniz's notation, and we like to use it still today. And do you know why? Because it works. Yes: treating the "differential quotient" as an actual quotient works.
For example, let's say that I have to calculate the derivative with respect to $t$ of
$$f(x(t))$$
How can I do it? Well, easy:
$$\frac{d f}{dt} = \frac{df}{dx} \frac{dx}{dt}$$
Or maybe I want to solve the differential equation
$$\frac{d y}{d x} = \frac{x^2}{y}$$
Piece of cake:
$$y \ dy = x^2 dx \to \int y \ dy = \int x^2 dx \to \frac{y^2}{2} = \frac{x^3}{3} + c \to y=\pm \sqrt{\frac 2 3 (x^3+c)}$$ 
And so on. In fact, there is even a theory in which this kind of notation is made rigorous: it is called non-standard analysis.
So, if you're a mathematician, then try to avoid using  $\frac{df}{dt}$. But if you are a physicist, then...go on, and have no fear!  
PS I almost forgot: if you are interested in the history of the concept of derivative, this article is really, really interesting.
A: In physics there are no infinitesimals, so dx is always treated as a "small but finite interval" during discussions, and only when the actual calculation is being done do we switch to mathematical mode, and "take the limit."
During the 17th and 18th centuries, mathematicians and physicists both did this all the time. As they say in sports "no harm, no foul!"  However, as analysis moved beyond its origins in physics, definitions and methods were tightened up, and proofs became more rigorous, including the definition of limits, and the methods required tend to overwhelm the physical content if included in a physics class.
So to recapitulate: mathematical rigor is always required in a mathematics, but physics is ultimately an experimental subject, and math may be the language of physics, but it is just a tool.
A: Using the word fraction is a more elementary way of naming the set of rational numbers : numbers that are represented by a ratio of integers. 
In physics when we model physical systems with differential equations we generally work in the domain of real numbers and sometimes complex numbers since we tend to think of real physical systems as existing in a continuum. Quantum mechanics may be one exception.
I can't name any physical system that might be modeled in the domain of rational numbers so I would have to say that $\frac{dy}{dx}$ is not a fraction.
