Question regarding differentiation of x with respect to x

You are using intuitive and imprecise notions. The derivative of a function f(x) is defined as a limit. Look at what the actual definition of derivative tells you about differentiating the function f(x) = x.

To some extent it is possible to reason intuitively with symbols like dx and dy, but it takes experience to know what sort of intuitive reasoning actually works.

 

If you increase [itex]x[/itex] by [itex]\delta x[/itex], then you have increased [itex]x[/itex] by [itex]\delta x[/itex].

 

It should be obvious that the "rate of change of x compared to the rate of change of x" is 1!

 

Think of it as rate of change of one thing w.r.t another.

So, if I get paid P=$2 per foot of ditch, and I dig S=3 feet per day, ##dP/dt = 2, dS/dP = 3##, and ##dP/dt = 6## (i.e., $6 per day). The last one, by the way, is the chain rule.

So, loosely speaking, ##dP/dP = 1## - i.e., I get one dollar for each dollar I get. And ##dS/dS## also equals 1 - I dig one foot for every foot I dig. Trivial and sloppy but true.

The notation is endlessly confusing, I would agree. Wait til you find yourself doing things like figuring the derviative of something with respect to another thing. For example, if## g = g(t), and f = f(q)##, and we then look at f(g), which we call f(g(t)), it gets mighty confusing to take the derivative of f w.r.t. g, ##df/dg.## if ##g = t^2##, you can find yourself taking ##df/dt^2##. In fact, if ##g(t) = e^{t^2}##, it is perfectly reasonable to find ##dg/de^{t^2}##