Newton vs Leibniz notation

Regarding the notations for the derivative:

Upsides of using Leibniz notation:

• It makes most consequences of the chain rule "intuitive". In particular, it is easier to see that $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$ than it is to see that $[f(g(x))]' = f'(g(x))\cdot g'(x)$. See also $u$-substitution, in which we "define $du := \frac{du}{dx}dx$".
• In a physical/scientific setting, it makes it obvious what the units of the new expression (integral or derivative) should be. For instance, if $s$ is in meters and $t$ is in seconds, clearly $\frac{ds}{dt}$ should be in meters/second.

Downsides:

• It is harder/clumsier to keep track of arguments of the derivative with this notation. For instance, I can more easily write and keep track of $f'(2)$ than I can $\left.\frac{dy}{dx} \right|_{x=2}$
• It often leads to the mistaken notion that $\frac{dy}{dx}$ is a ratio

Notably, almost no one uses Newton's notation for the integral ("antiderivative"), in which the antiderivative of $x(t)$ is $\bar x(t)$, $\overset{|}{x}(t)$, or $X(t)$ (though this last one occasionally is used in introductory textbooks). Leibniz notation seems to be the clear winner in that regard.

The most obvious difference is that the Leibnitz notation strictly defines what the independent variable is. In basic calculus we tend, as a rule, to derive a function "y" of a variable "x", but what happens when you want to derive the function $w = 3x+4m$? How would the Newton notation help you understand which is the variable and which is the parameter?

Also, in integrals, the notation makes methods like substitution or integration by parts much simpler as you use the "dx" symbol as if it were a substitutable variable.

I think it's best to use both notations simultaneously.

For instance, my preferred statement of the chain rule is:

$$\frac{d}{dx}f(y) = f'(y)\frac{d}{dx} y$$

For example, we can write:

$$\frac{d}{dx} \sin(x^3) = \sin'(x^3)\frac{d}{dx}x^3 = \cos(x^3)\cdot 3x^2 = 3x^2 \cos(x^3)$$

Try to do this using just Newtonian notation, or just Leibnizian notation; you'll quickly notice that both are harder.

There's also multivariable versions. For instance:

$$\frac{d}{dt}f(x,y) = (D_0 f)(x,y) \frac{d}{dt} x+(D_1 f)(x,y) \frac{d}{dt} y$$