Can the third derivative tell me anything about the original function?

The first derivative can tell me about the intervals of increase/decrease for $f(x)$. The second derivative can tell me about the concavity of $f(x)$.

So can the third derivatives, and any derivatives beyond, yield any useful piece of information for graphing the original function?

• Useful information about the function, or information helpful for graphing the function? – GFauxPas Apr 1 '15 at 23:42
• Wouldn't useful information about the function also be helpful for graphing the function? From what I've learned so far, I can draw most graph just using up to the second derivative, but I'm curious about the third derivatives and beyond. If the two you mentioned aren't the same, what's the different between them? – L to the V Apr 1 '15 at 23:45
• Technically, yes, but actually drawing carefully enough to demonstrate this effect is not easy. Hell, drawing carefully enough to demonstrate the effect of concavity is often not easy. – Ian Apr 1 '15 at 23:45
• I'm not sure what you're talking about. Can you elaborate a bit more? – L to the V Apr 1 '15 at 23:46
• I feel like the sequence of (ie.) first derivative is positive, second derivative is positive, third derivative is negative may tell you something about the coefficients of the best fit cubic polynomial at some particular point. – Matthew Levy Apr 1 '15 at 23:47

Through supremely literal, I guess just as we can think of the first derivative as 'how quickly the function changes' and the second as 'how quickly the function of how quickly the function changes changes', we can say that the third is just 'how quickly the function of how quickly the function of how quickly the function changes changes changes'.

I would say that concavity and slope only seem significant because we gave them very visual names. We gave them names because they were used in visualising functions, but they are nothing more than a geometrical interpretation of the above quoted statements.

If we insert the names of the previous iterations into those statements, we get that the second derivative is 'how quickly the slope changes' and the third derivative is 'how quickly the function of how quickly the slope changes changes', which is just 'how quickly the concavity changes'.

Maybe we can give this one a nice geometrical name too, something like 'flexion', because it's describing whether the concavity is becoming tighter or looser and how quickly it is happening.

But there's nothing particularly remarkable about this property, just as there is nothing inherently remarkable about slope or curvature (just that we have given them names).

• I like that a lot, how fast the concavity is changing. So how fast the graph of the function is pinching closer or widening up. – Matthew Levy Apr 2 '15 at 0:20
• So the 3rd derivative is a measure of how fast acceleration is changing, same as how acceleration is a measure of how fast speed changes. – L to the V Apr 2 '15 at 0:23
• Also note that the rate of change of acceleration is sometimes called jerk. – Paddling Ghost Apr 2 '15 at 0:29

Yes: it tells you about the rate of change of the curvature of a plane curve, which is given by the formula $$\kappa = \frac{y''}{(1+y'^2)^{3/2}}$$ The derivative of this is $$\kappa' = \frac{y'''}{(1+y'^2)^{3/2}} - \frac{3y' y''^2}{(1+y'^2)^{5/2}}.$$

If you work in more than two dimensions, the torsion of a curve involves the third derivative: this tells you how non-planar it is (the helix has non-zero torsion, for example).

• This interpretation works if y'=0 -- the (corrected) formula for the derivative of curvature in that case reduces to just y''', i.e., the jerk IS the derivative of curvature. But if y' is nonzero, then the connection between curvature and the second derivative becomes problematic. For instance, with y=e^x, the second derivative is exponentially growing while the curvature very quickly drops to zero. So this interpretation is probably best to ignore unless the tangent line is horizontal. – Barry Smith Jun 15 '17 at 15:32

It all depends on the function itself, because a linear function for example isn't concave in the first place. If you had a vector field represented by vector function "e.g. wind velocity", then you want to specify, are you talking about gradients or divergence or curl etc, as these are all derivation operators.

It might not be as useful as simply the second derivative.

But let's say you have a scalar field representing the elevation of a terrain, and you want to know how rough a terrain represented by 3d function, you can get an idea how rough it is from the third derivative. In a rough terrain, you will have a non-monotonic third derivative. Note, that this interpretation is strictly a creative way to look at it, as i don't have any reference supporting the previous.

When in college, I was told that the order of the derivative is related to the "dimension."

I think this is easier to see once you take integration. Double integrals give us regions in the two dimensions while triple integrals give us regions in three dimensions.