In one of the letters written to the Infeld group in Warsaw Einstein wrote:
"A new manner of thinking is essential if humankind is to survive."

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CONTEXT
The puzzle below is one of the series destined to show the inadequacy of the
established education. I have presented it to several hundreds of mathematicians
and physicists and got 2 correct answers, which speaks for itself.
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PUZZLE

Let y = f(x) derivable function of real x.
Let f'(x) derivative of f(x) with respect to x.
 
We shall call dy "differential" of y and write it:

dy = f'(x) dx
 
Question 1: Is dy finite or infinitesimal?
Question 2: What is its nature?
 
By "nature" we mean, for want of a better word, the "type" of mathematical 
concept, eg. vector, variable, number, space, increment, integral, geodesic, 
etc.                                                             
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TO THE READER
If you want to have fun, try to answer the questions by yourself, before looking
at the solution.

Let's note that the understanding of the nature of differential is necessary
to get the gist of Relativity. Indeed, its derivation and its fundamental
concepts are based upon the notion of differential.
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THE USUAL ANSWERS
Most of interviewed people answered:
1.Differential dy is infinitesimal
2.Differential dy is an increment

We shall comment these answers in the following order:
1."Infinitesimal"
2.Limit
3.Derivative
4.Differential
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INFINITESIMAL
Differential dy is NOT infinitesimal for the simple reason that there ain't no
such animal as "infinitesimal" in mathematics. Quite a shock to those who
learned maths in established schools and from established handbooks.
But shock or not, let's have a closer look at the thing. Infinitesimal is usually
defined as a number 
1.smaller than any positive real number, yet
2.greater in absolute value than zero.
Now, that is a glaring contradiction:
1. says clearly that "infinitesimal" is zero while 2. says clearly that it is 
not zero. Thus there is no such thing.

Question arises, why the overwhelming majority of handbooks is full of this 
phantom? It's determined by "history", or rather by ignorant, parrot-like
repetition of obsolete and erroneous historical terms.

It's true that Newton and Leibnitz used "infinitesimal" in the definition of
derivative. But THEY had an excuse, supposing that they need one. Having done
the deepest breakthrough in mathematics, they left the stable land on things
being this and that, for a swamp where everything moves, tends towards and
reaches without ever reaching. Apparent swamp, as we know it today, because we
know with help of topology and rigorous definition of limit that the concepts of
calculus are just as steady and stable as the rest of maths.
But they did not have those tools and, in order not to drown in moving sands
created intuitive crutches such as the "infinitesimal". 

THEY would have an excuse, if they needed one. But we have none for repeating 
like cuckoo clocks their crutch which became an obsolete, meaningless drivel.
 
The word "infinitesimal" is used in Physics in a different meaning of "small
enough to be approximated ...". Thus Einstein talks often about segments of
curves short enough to be approximated with segments of straight line, calling
them "infinitesimal". We find it unfortunate, but a genius may be improper in
metalanguage, impatient to move to formulas, where he is more at home and
more rigorous. However, we do not recognize for ourselves the right to repeat
his improper terms and prefer to call such segments "elementary". 
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LIMIT
In definition of limit the standard courses insist on necessity of using 
"infinitesimal", which muddles the definition making it on the one hand wrong 
and, on the other hand impossible to follow for a student, which makes him lose 
confidence in his own intelligence.

Here comes a proper definition:

Let a point A of Rn or Cn and an application f taking its values in Rp or Cp 
such that f is defined in a neighborhood U of A with possible exception of A. 
We say that f(x) TENDS TOWARDS B WHEN x TENDS TOWARDS A if, whatever the 
neighborhood V of B, there is a neighborhood U of A such that for every x 
distinct from A in U, f(x) belongs to V.
B is called LIMIT of f(x) and is noted:
B = lim f(x) (x-->A)  or
f(x) --> B when x --> A
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DERIVATIVE
Standard courses define derivative of y=f(x) as limit of D(y)/D(x) for D(x) 
tending towards zero. That, of course tends towards 0/0 which is meaningless, 
but the teacher replaces on the way D's by d's and declares of the blue sky 
that f'(x)=d(y)/d(x) where d(x) is the "infinitesimal" value of D(x) reached on 
the way towards zero and dy the corresponding "infinitesimal" value of D(y).

After having thus screwed up the derivative, our teacher multiplies illegally
the above "formula" by dx and declares that 
differential dy = f'(x) dx, where dy and dx are "infinitesimal" increments.

Illegally, because following his logic he multiplies illegal f'(x)=0/0 by 0 
and gets 0 = f'(x) * 0.

Now, the proper definition of derivative is:

Let f(x) analytical derivable function of real x.
 
Let p=(1/h)(f(xo+h)-f(xo) increase rate of f(x) at x=xo.
 
Then we define the derivative f'(x) at xo as:

(f'(x))(xo) = lim p (h->0)
 
As you see we do not have any DY/DX, nor it's limit for DX->0 being 0/0 , nor 
the whole screw up of DX->0 = ("infinitesimal")dx, nor that last dx confused 
with the differential "dx".
 
Free of these screw ups we may proceed towards the differential.
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DIFFERENTIAL
Given derivable function f(x) having derivative f'(x), we shall call 
"differential" a linear function allowing to pass from a number h to a number  
k = hf'(x) = df(h)
This function will be noted df or dy and dx.
In particular for dx: k = hx'(x) = h, which is identity function.
Therefore we may write:

k = dy(h) = f'(x)h = f'(x)dx(h)

and, finally, as equality of functions (independently of h):

dy = f'(x)dx
 
We confuse still quite often the functions dx, dy with the values they take 
dx(h), dy(h) when applied to a number h and we say improperly "given the 
increment dx of the variable x ..." instead of considering a number h which 
will engender a number k = dy(h).

As dy and dx are proportional to h, their ratio has constant value dy/dx = f'(x),
which justifies the notation dy/dx of the derivative.
 
CONCLUSIONS
 
1.The nature of dy is FUNCTION.
  This function associates with any (finite) h a (finite) k = hf'(x).
  For x=xo  k(xo) = h(f'(x))(xo), which is the function of the parallel to 
  the tangent to f(x) at xo passing par x=y=0.

2.It is neither finite nor infinitesimal:
  It is not infinitesimal because THERE AIN'T NO SUCH ANIMAL,
  It is not finite because function as such has no value.
  For a (finite) h it takes at xo the (finite) value k(xo) = h(f'(x))(xo).
  For instance, for f(x) = x**2, f'(x) = 2x.
  For x=5    f'(x)(5) = 2*5=10.
  For h=3 dx(3)=3 dy(3)=30, etc.   
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