What is a differential (Calculus) The differential of a function $y(x)$ is defined as, 
$$dy = f'(x)dx$$
I didn't know that a differential is actually defined by the above equation and is a function of both $x$ and $dx$, but does (1) the motivation comes from the linearization of a function?
(2) Is it mathematically valid to multiply and divide by differentials?
$$\frac{dy}{dx} = \frac{dy}{dx}, \text{then} \; dy = \frac{dy}{dx}dx$$
If it's not, then can we do,
$$\Delta y/\Delta x = \Delta y/\Delta x, \text{then} \;\Delta y = \frac{\Delta y}{\Delta x} \Delta x $$
And take the limit as $\Delta x \rightarrow 0, \Delta y \rightarrow 0$? So the end result is the same as if we just multiplied by differentials?
(3) Can a differential be defined by $\Delta x = dx$ when $\Delta x \rightarrow 0$?
(4) Is there a difference between writing,
$$\frac{d}{dx} y \; \text{and} \; \frac{dy}{dx}$$
Where the first is a derivative and the second is a ratio of partials?
 A: $\displaystyle\frac{dy}{dx}=lim_{\Delta x \rightarrow 0} \frac{\Delta y}{\Delta x}$. Hence, $\displaystyle\frac{dy}{dx}$ is a notation for this limit. Do not take it as a division. 
Regarding (4), both are same but the second is not a division as I said.
A: Yes, the differential is the best linear approximation of a function in a neighborhood of a point $a$. To be more specific, we would like to find a linear function $df_a$ such that $f(a+\Delta x)-f(a) \approx df_a (\Delta x) $, i.e. the difference in $f$ will be linearly related to the change in $x$. 
To be more precise, the differential of a function $f(x)$ at the point $a$ is a linear function $df_a (\Delta x) $ such that:
$$f(a+\Delta x)=f(a)+ df_a(\Delta x) + o(\Delta x)_{\Delta x\to 0} $$
The differential is unique and from this definition and the definition of the derivative, $df_a(\Delta x) = f'(a)\cdot \Delta x$. Following this, $\frac{df_a (\Delta x)}{\Delta x} = f'(a)$, and another way of thinking about it is $\frac{dy}{dx}=f'(x)$.
