# Significance of the third derivative

## Main Question or Discussion Point

I've been searching for some indication of the "significance" of higher order derivatives, but without much luck. Perhaps someone here can offer some insight.

The first derivative of a function f(x) (with respect to x) gives the slope of the tangent to the curve of that function. When this derivative is 0, the function is at a local extremum.

The second derivative of a function gives the concavity, and when this is 0, it is a critical point of the function.

What would the analog for third derivative be, (and for higher order derivatives)?
Or is it perhaps the case that our brains are not prepared to fathom their significance?

BiP

I've been searching for some indication of the "significance" of higher order derivatives, but without much luck. Perhaps someone here can offer some insight.

The first derivative of a function f(x) (with respect to x) gives the slope of the tangent to the curve of that function. When this derivative is 0, the function is at a local extremum.

The second derivative of a function gives the concavity, and when this is 0, it is a critical point of the function.

What would the analog for third derivative be, (and for higher order derivatives)?
Or is it perhaps the case that our brains are not prepared to fathom their significance?

BiP
If you're looking for physical significance, intuition soon fails. The first derivative of position is velocity; the second derivative is acceleration; the third derivative is the instantaneous change in acceleration; etc. It's hard to visualize.

But there's a mathematical way to look at this. Say I want to approximate an arbitrary function by a polynomial.

If the given function happens to be suitably well behaved, it will have derivatives of all orders. In that case we can approximate the function with a polynomial by taking the first n terms of the Taylor series.

The n-th term of its Taylor series involves the n-th derivative of the function. So each additional derivative gives us one more term of the polynomial to get a slightly better fit.

The Wikipedia page has a nice graphic (halfway down the page) showing how as you add one more term of the Taylor series for ex, you get a better and better fit.

http://en.wikipedia.org/wiki/Exponential_function

So the answer to your question is that the n-th derivative gives an additional correction factor when approximating a given function by its Taylor series. And as you let the number of terms go to infinity, the polynomial becomes an "infinite polynomial," which we call a power series, that is exactly equal to the original function.

You could say that all of the the n-th derivatives encode information about a gven function. The more derivatives, the more information you have.

In a sense, all (suitably well behaved) functions are just infinite sums of polynomials. And the n-th term of that infinite sum is determined by the n-th derivative. So that's what all those derivatives are really for! To break down an arbitrary function into simpler pieces that are easier to understand and work with.

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If you're looking for physical significance, intuition soon fails. The first derivative of position is velocity; the second derivative is acceleration; the third derivative is the instantaneous change in acceleration; etc. It's hard to visualize.

But there's a mathematical way to look at this. Say I want to approximate an arbitrary function by a polynomial.

If the given function happens to be suitably well behaved, it will have derivatives of all orders. In that case we can approximate the function with a polynomial by taking the first n terms of the Taylor series.

The n-th term of its Taylor series involves the n-th derivative of the function. So each additional derivative gives us one more term of the polynomial to get a slightly better fit.

The Wikipedia page has a nice graphic (halfway down the page) showing how as you add one more term of the Taylor series for ex, you get a better and better fit.

http://en.wikipedia.org/wiki/Exponential_function

So the answer to your question is that the n-th derivative gives an additional correction factor when approximating a given function by its Taylor series. And as you let the number of terms go to infinity, the polynomial becomes an "infinite polynomial," which we call a power series, that is exactly equal to the original function.

You could say that all of the the n-th derivatives encode information about a gven function. The more derivatives, the more information you have.

In a sense, all (suitably well behaved) functions are just infinite sums of polynomials. And the n-th term of that infinite sum is determined by the n-th derivative. So that's what all those derivatives are really for! To break down an arbitrary function into simpler pieces that are easier to understand and work with.
I see. I know that the n-th derivatives are used to generate the coefficients of the Taylor polynomial of a given function. I was just wondering about the intuition behind the third derivative in the same way the second derivative gives an intuition about concavity etc.

BiP

According to Wikipedia, the third derivative of position with respect to time is also called the "jerk". You experience a jerk when there is a sudden change in acceleration (your seat suddenly pushes much harder on you in a car, for example, and the jerk measures how fast that force is changing). A large jerk on your body can cause a physiological impact.

Analogously, the fourth derivative of position with respect to time is also called the "jounce".
I don't know of many applications of this, although the Euler beam equation uses 4-th order derivatives, but with respect to distance, not time. (I don't know much about that equation, but it is related to bending of solid structures).

http://en.wikipedia.org/wiki/Jerk_(physics)

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I'm not asking about applications of the third derivative in physics. I'm asking about how the third derivative can be visualized by the graph of f(x) such as how first derivative can be understood by tangent lines and second derivative can be understood by concavity.

BiP

I'm not asking about applications of the third derivative in physics. I'm asking about how the third derivative can be visualized by the graph of f(x) such as how first derivative can be understood by tangent lines and second derivative can be understood by concavity.
Here's a few ways to think about it. If the first derivative is positive at x=x0, then that means that if you approximated f(x) with a line y=ax+b passing through x=x0, a would be positive, and we call that "increasing" (locally). If the second derivative is positive at x=x0, then if you approximated f(x) with a parabola y=ax^2 + bx + c passing through x=x0, then a would be positive, and we call that "concave up". If the third derivative is positive at x=x0, then if you approximate f(x) with the cubic function y=ax^3 + bx^2 + cx + d passing through x=x0, then a would be positive, and we call that ... unfortunately, we don't have a name for that. But you can see for yourself what it means for the leading coefficient of a cubic function to be positive.

Also, if the third derivative is positive, that means that the second derivative is increasing, which means that the first derivative is concave up. In other words, the slope of the original graph increases faster than just a constant rate of increase. So the slope is becoming steeper at an accelerating rate!

I hope that helps.

This is the kind of answer I was looking for. Thank you lugita. This is insightful indeed. I might make a name for it some day if I become a mathematician!

BiP

As was said above... When the back of your chair (while driving) is exerting a certain "pressure" on your back, acceleration is constant. When that pressure increases or decreases, you are experiencing jerk.

One way to look at seeing it on a graph, is the deviation in a graph from being parabolic, and you know what a parabola looks like, however it might be hard to eyeball that. A cubic or quartic is hard to differentiate from a parabola in places.