I was reading in calculus 3 book , but I am stuck here
I don't understand what $\frac{\partial}{\partial x}$ means (I know it is partial derivative but I don't know what it means without f like $\frac{\partial f}{\partial x}$ ) and I don't understand how does $\frac{\partial}{\partial x}*Q=\frac{\partial Q}{\partial x}$ and what does $\nabla $ means without any function ? It is like multiplying a number with a sign, which doesn't make any sense I heard from 3b1b video that this is not only a notional trick but there is a relation between curl and cross product but I didn't understand what he means it is in this video at 11:00
$\frac{\partial}{\partial x}$ is an operator, or an operation, which is a kind of function, that maps (differentiable multivariate) functions to other functions.
You're already familiar with other operators, for example "$+$" is an operator between pairs of numbers and numbers - we could just as easily write $+: \mathbb{R}^2 \rightarrow \mathbb{R}$ and use the notation $+(a, b)$, e.g. $+(3, 4) = 7$, but in practice we put the $+$ between the values instead.
So without a function being involved, $\frac{\partial}{\partial x}$ is just an operator floating around waiting to be applied to a function, and putting it to the left of a function is how we apply it, i.e. $\frac{\partial}{\partial x} f = \frac{\partial f}{\partial x}$ (where on the left we have "differential operator being applied to a function", and on the right we have "derivative of a function").
In this form, we can manipulate it algebraically to some extent. For example, $\frac{\partial}{\partial x} + \frac{\partial}{\partial y}$ is an operator, and it works so that if we apply it to a function $f$ it behaves in a sensible way, that is $\left(\frac{\partial}{\partial x} + \frac{\partial}{\partial y}\right)f = \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y}$. We can even give it a name, so $\nabla = \frac{\partial}{\partial x} + \frac{\partial}{\partial y}$.
This isn't so different to being able to write $f$ on its own without asking "$f$ of what?". When you write $f + g$ you know that this represents "the function that applies $f$ and $g$ separately and then adds the results", which makes sense even when you don't write it explicitly as $(f + g)(x) = f(x) + g(x)$, and these operators take that abstraction one step further.
Expressing the curl as a matrix determinant is a little weirder, but it's mostly just taking advantage of the fact that the expression happens to look just like the $3 \times 3$ determinant. You could do full-on matrix algebra with operators, but you have to be careful with some things that are natural for numbers but not functions (For example, what's $\left(\frac{\partial}{\partial x}\right)^{-1}$? Some kind of integral?)