Operations with $\frac{dy}{dx}$. While working on a problem I have found a solution. I am curious about a clean and correct way to write it down. I want to find the derivative of $y(x)=:y$.
$$(5y^4+1)\frac{dy}{dx} + 1 = 0\\ \frac{dy}{dx} = -\frac{1}{5y^4+1}$$
Is it mathematically correct to divide by $(5y^4+1)$? I am aware that it is greater than zero, however is it really multiplied with $\frac{dy}{dx}$? Is it an operator? What is the correct way to write it down?

Yes, what you're doing is perfectly "legal". Note that $\frac{dy}{dx}$ is the differentiation operation $\frac{d}{dx}$ being applied to $y$, and is thus the result of applying an operator; which is a function. Meanwhile $\frac{d}{dx}$ on its own is the differential operator, which maps functions to functions.
