The derivative (gradient) of gravitational acceleration with respect to distance is what gives rise to tides.

The derivative of gravitational acceleration with respect to time is what gives rise to jerk.

For an object in free-fall in a gravitational field, the distance changes periodically with time. So I suggest that tides are the most obvious indication of jerk. Within the Roche limit, disintegration is an indication of jerk; spaghettification is the equivalent near a black hole.