This section deals with straightforward derivatives of functions of time. It is related to the sections on
Functions of time
We use functions of time, $f(t),$ to describe many important quantities, such as
- Position of an object over time (velocity)
- Velocity of an object over time (acceleration)
- Account balances over time
- Business inventory change over time
- Change in flow volume over time
We often use the word dynamic to describe systems that change over time, as opposed to static systems. Derivatives of quantities with respect to time is how we analyze dynamic systems. The time derivative lets us find instantaneous values of a time-dependent quantity rather than averages. Let's use position and a position function to illustrate the idea.
Position, velocity, acceleration, ...
Let's imagine that we toss a ball straight into the air near the surface of Earth. It's height over time is specified by the time-based function
$$h(t) = h_o + v_o t + \frac{1}{2} gt^2,$$
where $h$ is the height, $h_o$ is the initial height (height from which the ball was thrown), $v_o$ is its initial velocity, $g$ is the acceleration of gravity on Earth (9.8 m/s2), and $t$ is the time in seconds.
Derivatives
the first derivative of $h(t)$ is
$$h'(t) = v_o + gt,$$
where $h_o$ is constant. The second derivative is derivative of h'(t),
$$h''(t) = g$$
The second derivative is just the constant acceleration, $g = 9.8 \frac{m}{s^2}.$
For any position function, the first and second derivatives are:
- x(t) = position function
- x'(t) = velocity (change in position over time)
- x''(t) = acceleration (change in velocity over time)
for a more complicated position function, such as one might encounter in describing the motion of a quad-copter (drone), there can be derivatives higher than the second. These can be useful in physics and engineering, and thus they have names:
- x'''(t) = jerk (change in acceleration over time)
- x(4)(t) = snap (change in jerk over time)
- x(5)(t) = crackle (change in snap over time)
- x(6)(t) = pop (change in crackle over time)*
*
Interpreting x(t) vs. t graphs
Here is a made-up graph of position, $x(t),$ vs. time $(t)$ for a "particle" moving in one dimension, in this case, along an x-axis. In has six segments from which we can learn how to read such graphs for clues about velocity and acceleration of the particle.
It's not uncommon in theoretical situations like this to refer to the moving body as a "particle." Just get used to it. It's just a generic object and the details don't matter.
x(t) vs. t for a particle
moving along the x-axis
Velocity is the first derivative of position, so instantaneous velocity is the slope of this graph at any point in time. If the velocity (slope) is constant over an interval, then the particle is not accelerating over that interval; the second derivative would be zero. If the x(t) vs. t graph is curved in an interval, then the particle is accelerating over that time.
This table describes the motion of the particle in each time interval shown on the graph.
| Interval | Motion |
|---|---|
| 0 → t1 | The particle moves along the x-axis in the +x direction at a constant velocity because the slope of the segment is constant. |
| t1 → t2 | The particle does not move during this interval, therefore its velocity is constant. |
| t2 → t3 | The particle moves in the +x direction during this interval, with increasing speed. That is, it accelerates. The 2nd derivative of the position in this interval is non-zero. |
| t3 → t4 | the particle does not move during this interval: v = 0. |
| t4 → t5 | The particle moves in the -x direction in this interval, and the speed with which it moves decreases — it decelerates*. |
| t5 → t6 | The particle moves in the -x direction in this interval with a constant velocity. |
* Deceleration is usually just called acceleration in physics and math, and just given a negative sign.
xaktly.com by Dr. Jeff Cruzan is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. © 2012-2025, Jeff Cruzan. All text and images on this website not specifically attributed to another source were created by me and I reserve all rights as to their use. Any opinions expressed on this website are entirely mine, and do not necessarily reflect the views of any of my employers. Please feel free to send any questions or comments to jeff.cruzan@verizon.net.