$$\frac{d}{dx}(\frac{6x^4+4x^2}{x})=18x^2+4\tag{1}$$

$$\frac{d}{dx}(6x^3+4x)=18x^2+4\tag{2}$$

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?

## Best Answer

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.