$$\frac{d}{dx}(\frac{6x^4+4x^2}{x})=18x^2+4\tag{1}$$
$$\frac{d}{dx}(6x^3+4x)=18x^2+4\tag{2}$$
According to this answer, (1) & (2) are different functions (see case 2). (1) & (2) have the same derivative. How can two different functions have the same derivative?
Assuming these are functions of a real variable and their domains are presumed to be the largest subsets of reals for which the functions are defined, the answer is: they actually do NOT have the same derivatives, because these derivatives have different domains (just as the original functions do).
In (2), the domains for both the function and its derivative is $\mathbb{R}$.
In (1), the domains for both the function and its derivative is $\mathbb{R}\setminus\{0\}$; note that it's the domain for the derivative because the derivative of a function wouldn't make any sense at the points where the original function isn't defined.
They are different functions, but there's only one reason for that: they have a different domain. One has the domain $\mathbb R$, while for the other it's $\mathbb R\backslash \{0\}$. But at all points where their domains overlap, they have the same value. So it shouldn't come as a surprise that their derivatives are the same where their domains overlap. The only difference between the derivative functions: they also have different domains. A derivative function can't have a larger domain than the function it's a derivative of!So to be precise, their derivatives are also not the ssame. One has domain $\mathbb R$, the other $\mathbb R\backslash\{0\}$. And they, too, have the same values where their domains overlap. But that's not enough for to two functions to be equal. Functions are only equal if their domains and the values they have are the same (and there codomains are equal, but that point is usually left out in lower level classes).
