Yes. The higher derivates are useful in many cases, both as sought properties and observations. The invariances implied by relativity,(the trivial notion that the universe behaves the same regardless of where you select the center), mean that most laws are defined on the second derivative. Taylor approximations are useful to approximate something locally, but properties of the system over wider regions generally need to account for the higher derivatives. You can see this in e.g. simulating a system over time requires that the derivatives at the borders of the valid taylor approximation region to be included as diracs.
Or in other words, you can approximate exp(x) as a set of first order taylor approximations that each covers a small window to arbitrary precision, but the combination of them is still has well defined higher derivatives that are not 0.
Or in other words, you can approximate exp(x) as a set of first order taylor approximations that each covers a small window to arbitrary precision, but the combination of them is still has well defined higher derivatives that are not 0.