What's the difference between d, d/dx and dx?

[Jason Ko]

TL;DR

What's the difference between d,d/dx and dx?

What's the difference between d,d/dx and dx?

 

[anuttarasammyak]

For an example of function $y = \sin x$.
$$dy=\cos x \ \ dx$$
dx is infinitesimal change of x and dy is change corresponding to dx.
$$\frac{d}{dx}\ y=\frac{dy}{dx}= \cos x$$
$\frac{d}{dx}$ is operator to make differential coefficient
which is expressed as the ratio $\frac{dy}{dx}$.

 

[PeroK]

Summary:: What's the difference between d,d/dx and dx?

What's the difference between d,d/dx and dx?

$d$ has no independent meaning in terms of calculus.

$d/dx$ denotes the derivative (of a function) with respect to $x$.

$dx$ denotes the differential of the variable $x$. Informally, you can think about it as a very small or infinitesimal change in $x$. More formally, it is as described in post #2 above and here, for example:

https://tutorial.math.lamar.edu/classes/calci/differentials.aspx

 

[Mark44]

d/dx denotes the derivative (of a function) with respect to x.

Your explanation doesn't distinguish between the action of taking a derivative, versus the derivative itself. I would say that $\frac d{dx}$ is the operator that when applied to a function, produces the derivative of the function with respect to x.
If f is the function, then $\frac {df}{dx}$ is the derivative of f with respect to x.