How to know if the derivative of a function has an asymptote This is my function $f(x)$![enter image description here]( What is the methodology for me determining if the sketch of the derivative has a horizontal asymptote? basically i want to be able to justify that it has a horizontal asymptote and not just because it looks like it does

By looking at your picture (with no values, or scale) it seems that for large positive $x$ and for large negative $x$ the slope is $0$. Thus, the value of the derivate for large positive $x$ and large negative $x$ is $0$. This suggest that the derivative of $f$, i.e $f'$ has the horizontal asymptot $y=0$.

Without additional info its hard to say much more. And it also could be false, since we don't know how much of $f$ is shown in the picture.