I can understand the significance of computing for a derivative $\frac{dy}{dx}$ because you can see the slope (change in y in relation to the change in x) at any given point of a function.
By that logic, computing for $dy$ can find the change in the y axis at a given point x? What is the point of this?
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In calculus, we generally define $\frac{d}{dx}$ as an operator that acts on a function $y$, meaning that $\frac{dy}{dx}$ is just the output of an operator and the top and bottom are not independently defined. Throughout history many people have had the same urge as you to defined "dy" and "dx" independently, which leads to a field known as infinitesimal calculus.
When Newton was first creating calculus, he originally thought of dy and dx as infinitesimals as well, and the math was only formalized with operators later.
Personally, I find taking the differential instead of the derivative makes it easier to keep my options open, and also makes it more amenable to multivariable equations.
For instance, say you have the equation $y^2 = e^x + z$. If you take the differential, you get $2y\,dy = e^x\,dx + dz$. From here, you can solve for the derivative $\frac{dy}{dx}$ if you like. Or, you can divide both sides by $dt$ and get related rates. You can set one of the differentials to zero to get partial differentials, etc.
Anyway, there's nothing wrong with the derivative, but, to me, it seems that always forcing it into a form of a derivative is messy.
Additionally, there are a few places where the derivative is misleading. Take the equation $x = 0$. The derivative with respect to $x$ would lead you to say $1 = 0$, but the differential is more clear: $dx = 0$.