About differential of functions... Ok, probably it is a silly question, but I'm studying calculus for the first time, and I still can't "see clearly" the definition of differential of a function.

We define $dy$ - or equivalently $df(x)$ - as $dy = f'(x)\cdot\Delta x$. Now, if we consider the function $y=f(x)=x$, it follows that $df(x)=dx=f'(x)\cdot\Delta x=1\cdot\Delta x=\Delta x$; hence we can rewrite the previous relation as $dy=f'(x)\cdot dx$; and consequently $f'(x)=\frac{dy}{dx}$.

The question is: is there any good reason why I should substitute $\Delta x$ with its differential $dx$ in that relation?

I know it might sound weird, but I can't intuitively "feel" the fact that $\Delta x = dx$, I think it just makes things more confused...

Source (lang: italian): angeloangeletti.it/UNICAM_RECANATI/APPUNTI/04_lez.pdf
Well, to understand this, you have to understand limits. What a derivative is doing is it is taking the slope of a polynomial function - f(x). What you are doing is you're taking two points exactly $\Delta$x apart. The points would then have x-coordinates x and x+$\Delta$x. When you take the slope of the line between those two points (Click this to help), you get a number, and when $\Delta$x is infinitely close to, but not equal to 0, we call it dx, and the slope of the line is the derivative at that point. Since we are getting numbers for points, we can plot the derivatives at all points and match them into a function. This function is the first derivative of your original polynomial function, f'(x). Taking the derivative of that function is the second derivative, f''(x). 

This is very useful because you can find, easily, rates of change, volume from outlines, and so on. For example, the derivative of a graph of the distance a car has traveled at a certain time is equal to the graph of its velocity at a certain time. 

The volume of a cone/pyramid, and therefore the volume of a sphere, are all derived by calculus. 

P.S. In the image, h = $\Delta$x
Well, to understand this, you have to understand limits. What a derivative is doing is it is taking the slope of a polynomial function - f(x). What you are doing is you're taking two points exactly $\Delta$x apart. The points would then have x-coordinates x and x+$\Delta$x. When you take the slope of the line between those two points (Click this to help), you get a number, and when $\Delta$x is infinitely close to, but not equal to 0, we call it dx, and the slope of the line is the derivative at that point. Since we are getting numbers for points, we can plot the derivatives at all points and match them into a function. This function is the first derivative of your original polynomial function, f'(x). Taking the derivative of that function is the second derivative, f''(x). 

This is very useful because you can find, easily, rates of change, volume from outlines, and so on. For example, the derivative of a graph of the distance a car has traveled at a certain time is equal to the graph of its velocity at a certain time. 

The volume of a cone/pyramid, and therefore the volume of a sphere, are all derived by calculus. 

P.S. In the image, h = $\Delta$x