Roughly speaking, the second derivative measures how the rate of change of a quantity is itself changing; for example, the second derivative of the position of a vehicle with respect to time is the instantaneous acceleration of the vehicle, or the rate at which the velocity of the vehicle is changing with respect to time.
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This slope therefore defines the average acceleration over the interval, and reducing the interval infinitesimally gives \ begin { matrix } \ frac { dv } { dt } \ end { matrix }, the instantaneous acceleration at time t, or the derivative of the velocity with respect to time ( or the second derivative of the position with respect to time ).
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If we make two measurements in a small enough time interval then we can * approximate * the instantaneous velocity of the object, and if we make more measurements in a small enough time interval then we can * approximate * its instantaneous acceleration, jerk etc . But I don't know of any way to actually * measure * instantaneous velocity, acceleration, jerk etc in a general reference frame.
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