Difference between this operators

Question

Which is the difference between this operators, acting over a mangitude $A(x,x',t)$?

a)

• $ \delta A/ \delta x$
• $dA/dx$

b)

• $ \delta A/ \delta x$
• $\partial A/\partial x$

often used is physics?

Answer 1

There is no real difference in either case.

Typically, one uses $\frac{\partial}{\partial x}$ when one thinks of $A$ as a function of multiple variables, with $x$ being merely one of those variables. On the other hand, one uses $\frac{d}{dx}$ when one thinks of $A$ as a function of a single variable, perhaps holding the other parts constant.

For instance, consider the polynomial $p(x, a, b) = ax + b$. Then $$ \frac{\partial p}{\partial x} = a$$ and $$ \frac{dp}{dx} = a.$$ But in the latter, we are thinking of $a$ and $b$ as being fixed constants.

It should be noted however that many authors ignore or abuse this convention, so it is necessary to infer specific meaning from context.