d/dx Notation Explanation please?

If you have a function $f$ with the independent variable $x$, then $$ \frac{d}{dx} f(x) $$ means the derivative of $f$ with respect to $x$. We also sometimes write this as $f'(x)$. Now if you have a function like $$ f(x) = ax, $$ then the derivative is $$ f'(x) = a. $$ This is clear because in writing $f(x)$ we have indicated that the function $f$ is a function of the variable $x$. If I instead had told you that $$ y = ax $$ and I just asked you to find the derivative, what would you do? You would probably, again, just say that the derivative is $a$. But in this situation it actually isn't clear what is a variable and what is a constant. And therefore we can write $$ \frac{dy}{dx} \quad\text{or}\quad \frac{d}{dx}y $$ to indicate that we are considering $y$ as a function of the variable $x$ and we are considering $a$ as a constant (In the case of multivariable functions we really should be taking about partial derivatives in this case). Now you could also write $$ \frac{dy}{da} $$ and in this case you are saying that $a$ is a variable. So the $\frac{d}{dx}$ notation is very helpful when you have expressions where there are several letters.

So what is $\frac{d}{dx}$? You can consider this as an operator that takes as "input" a (differentiable) function and "outputs" a function.

$\dfrac{dy}{dx} = \dfrac{d(y)}{dx} = \dfrac{d}{dx}(y)$.

$\dfrac{d}{dx}$ is the differential operator. It tells you what operation (differentiation) you are doing and with respect to what variable. $\dfrac{dy}{dx}$ is the actual derivative of a function $y$ with respect to $x$. The operator can be applied to any function, for example $\dfrac{d}{dx}(x^2) = 2x$. If we wrote the same thing with $\dfrac{dy}{dx}$, then we get $\dfrac{dy}{dx}(x^2) = x^2\dfrac{dy}{dx} = x^2y'$. Since $\dfrac{dy}{dx}$ is already differentiating the function $y$, it does not do anything to $x^2$. The former example shows the differential operator being applied to a function. The latter shows the derivative of $y$ being multiplied by a function. The $x^2y'$ expression is often used in differential equations as shorthand to replace the longer to write $\dfrac{dy}{dx}$.