For a function \(f,\) the function \(f'\) denotes the first-order derivative of \(f,\) which represents the instantaneous rate-of-change of \(f\) at a point. We can take the derivative of a derivative though. The function denoted \[ f'' \qquad\text{or}\qquad (f')' \qquad\text{or}\qquad \frac{\mathrm{d}}{\mathrm{d}x} \Biggl( \frac{\mathrm{d}}{\mathrm{d}x} \biggl( f \biggr) \Biggr) \qquad\text{or}\qquad \frac{\mathrm{d^2}}{\mathrm{d}x^2} \biggl( f \biggr) \] is the second-order derivative of \(f,\) the derivative of the derivative, and represents an instantaneous rate-of-rate-of-change of \(f\) at a point. E.g. if \(f\) represents a position and \(f'\) represents a velocity, then \(f''\) represents the rate-of-change of the velocity, the acceleration. Additionally, the function denoted \[ f''' \qquad\text{or}\qquad \bigl((f')'\bigr)' \qquad\text{or}\qquad \frac{\mathrm{d}}{\mathrm{d}x} \Biggl( \frac{\mathrm{d}}{\mathrm{d}x} \Biggl( \frac{\mathrm{d}}{\mathrm{d}x} \biggl( f \biggr) \Biggr) \Biggr) \qquad\text{or}\qquad \frac{\mathrm{d^3}}{\mathrm{d}x^3} \biggl( f \biggr) \] is the third-order derivative of \(f,\) the derivative of the derivative of the derivative, and represents an instantaneous rate-of-rate-of-rate-change of \(f\) at a point. E.g. if \(f''\) represents an acceleration, then if \(f'''\) represents the rate-of-change of the acceleration, the jerk.
This can be continued indefinitely, the derivative of each order representing the rate-of-change of the derivative of the order before it. Adding more and more prime marks becomes unwieldy, so if we must refer to it we denote the \(n\)th-order derivative of \(f\) as \(f^{(n)}.\)