As for my knowledge there are 3 different notations for showing derivative for $f:C\to S\left(C,S\subseteq\Bbb R\right)$:
$f'$-Lagrange's way
$\dfrac{df}{dx}$ - Leibniz's way
$\dot{f}$ - Newton's way
my question is:
is there a fundamental difference between the 3 ways? for example i know that Newton's notation is mainly used in physics in respect to time. does it mean that this way gives a different result from the 2 others? or maybe Newton talked about it in a function that is in respect to time the first time and it like this today solely because of historical reasons?
These 3 different notations are notations of exactly the same thing, the derivative. It is customary in physics and engineering to denote a spatial derivative in Lagrange's way, and a time derivative in Newton's way, but there is nothing deep into it, it is just customary.