What is an example of an application of a higher order derivative ($y^{(n)}$, $n\geq 4$)?

If $y(t)$ denotes the position at time $t$, then:

• The first derivative, $y'(t)$, denotes velocity at time $t$.
• The second derivative, $y''(t)$, denotes acceleration at time $t$.
• The third derivative, $y'''(t)$, denotes the jerk or jolt at time $t$, an important quantity in engineering and motion control
• The fourth derivative, $y^{(4)}(t)$, denotes the jounce at time $t$; the jounce is also used in studying motion, and in studying the cosmological equation of state.

Fifth and sixth derivatives of position are also important in some applications/theoretical physics studies, but they have no universally accepted name.

You can also see this article from the Proceedings of the National Academy of Sciences discussing the use of the fifth derivative and curve fitting to do DNA analysis and population matching.

And higher derivatives are also used for approximating functions using Taylor polynomials, which can be useful when a certain amount of precision is required.

The Euler-Bernoulli equation, which describes the relationship between a beam's deflection and the applied load, involves a 4th derivative.

For an everyday answer instead of a mathematical formula: when you drive a car, the position of the steering wheel and of the gas pedal determine the acceleration of the car as a whole (that is, the second derivative of the car's position). Therefore when either the steering wheel or the gas pedal undergoes acceleration, that acceleration translates into the fourth derivative of the car's position.