Higher Order Derivatives and Their Applications (Explaining Calculus #12)

Up to this point, I’ve focused my efforts on derivatives of functions and what those derivatives mean. In particular, derivatives tell us about how things change over time – with derivatives, we can measure quantities like speed and growth. But derivatives are also functions, which means they have their own derivatives. Can we learn anything from there “derivatives of derivatives”? The answer is yes – and this is the topic for discussion here.

What are Higher Derivatives?

Recall briefly the definition of a derivative – for any function f(x), its derivative f^\prime(x) is defined by

f^\prime(x) = \lim\limits_{h \to 0} \dfrac{f(x+h) - f(x)}{h}.

I won’t actually need to use this definition here. But it is important for this discussion to remember that, in the first place, a derivative is a certain type of function. We begin with a function f(x), and we applying the limiting process above to f(x) to obtain the derivative function f^\prime(x).

The second-order derivative of the function f(x), usually just called the second derivative of f(x), can be found be applying this process twice – that is, applying it to f(x) to get f^\prime(x), then applying the same process again to f^\prime(x) to get another function, which we call f^{\prime \prime}(x) (two of the \prime because you do the derivative twice). This is called the second derivative. If you take even more derivatives, we generally call those higher derivatives.

How to Calculate Second Derivatives

We can use the formula for derivatives to find second derivatives. Since the second derivative f^{\prime \prime}(x) is the derivative of the derivative – we can apply the definition of the derivative to f^\prime(x) instead of to f(x). So,

f^{\prime \prime}(x) = \lim_{h \to 0} \dfrac{f^\prime(x+h) - f^\prime(x)}{h}.

We can briefly calculate a few second derivatives to show how this works. For a fairly simple one, set f(x) = x^3 + 2x. Using the “power rule” for derivatives, the first derivative of f(x) is f^\prime(x) = 3x^2 + 2. By applying the process again, we can find the second derivative, f^{\prime\prime}(x) = 6x. As another example, we could choose the more complicated g(x) = \ln{x}. In this case, the first derivative is g^\prime(x) = \dfrac{1}{x}. We can then use the power rule to find the second derivative, g^{\prime\prime}(x) = \dfrac{-1}{x^2}.

The Meaning of the Second Derivative

Now, for the purposes of my writing here, I don’t care that much about calculating lots of second derivatives. I’m much more concerned with what sorts of things you can do with second derivatives. Just as the first derivative has built into it the ability to tell us how to maximize and minimize things and how to calculate speeds, the second derivative also carries very useful built-in information. I’ll list out a few of these.

Application #1: Concavity

The most direct piece of information you can learn from second derivatives is called the concavity of a graph. Concavity describes the way a graph bends. As an example, the U-shape “bends upward” as you trace along the figure, and we call that “concave up”. When you work out the details, you can convince yourself that whenever f^{\prime\prime}(x) > 0, the graph of f(x) is concave up. This is because the idea of “bending upward” can be translated as” the slope is getting steeper and stepper in the up direction”. This second way of phrasing it talks about a slope (think first derivative) changing over time (so take another derivative). In much the same way, an upside-down U-shape has the visual feature of “bending downwards,” and in math we call that “concave down”. You can follow along the same lines of reasoning as with “concave up” graphs that if f^{\prime\prime}(x) < 0 then the graph of f(x) is concave down.

Application #2: Finding Max/Min Values

We can make use of this concavity idea to devise an entirely new method for finding max and min values of graphs. To explain this, let’s first give a brief reminder of how the first derivative enabled us to do this.

If we find the peak of a graph, that peak if definitely a so-called critical point – which just means f^\prime(x) = 0 there. Intuitively, this is like saying the top of a mountain is either flat or a sharp point, because if it were sloped then it wouldn’t be the top. So, if we want to know the max and min values of f(x), the first thing we can do is to solve the equation f^\prime(x) = 0. Once we find the solutions, we can then ask ourselves what happens to f^\prime(x) near those points. If, for example, we study the values of f^\prime(x) and find that the graph is going up right before it gets to a certain point, then becomes flat, then starts going downward after we pass by that point, then we can visualize this as a peak in the graph. If instead the graph was going down towards a point, then flattened out, then started climbing upward, then we’ve found a valley rather than a peak.

The second derivative gives us a more straightforward way to handle the “going up/going down” aspect of this problem. Take, for example, the case of finding a peak. The description “going up, flatten, going down” that I gave describes an upside-down U-shape. But we now know that second derivatives tell us whether a graph has an upside-down U-shape! If we know f^{\prime\prime}(x) < 0 then this is the same thing as the graph having an upside down U-shape. So, instead of doing the tedious work of evaluating lots of first derivative values nearby the critical point we care about – we can just evaluate f^{\prime\prime}(x) at the actual critical point itself and we can get the same information. In the exact same way, if f^{\prime\prime}(x) > 0 at the flat point on the graph, then you get a normal U-shape and you have a valley in your graph.

So, in almost every situation, using the second derivative actually makes finding peaks and valleys easier. The only hitch is that if you get unlucky and f^{\prime\prime}(x) = 0 exactly, then you can’t make things out one way or the other. If that unlucky event happens, you would need to go back to the old trick. But this is a very rare situation. There are technical reasons why the zero points of f^\prime(x) and f^{\prime\prime}(x) are nearly always different from one another. (If you want to challenge yourself, pick some random equations of the shape f(x) = x^3 + ax + b + c and see if you can find one where f^\prime(x) and f^{\prime\prime}(x) have a zero at the same place. You likely won’t find one by randomly guessing – you’ll have to be a bit clever. You’ll need to make use of an idea called “multiple zeros” or “repeated zeros”.)

Application #3: Acceleration and Gravity

In speaking of why derivatives are so important, I spoke about how you can use a derivative to measure speed – how fast something is going. I would now like to extend this interpretation of derivatives to acceleration, and to show how this interpretation contributed to one of the most important developments in the history of science.

We recall that the derivative of something is a description of how that thing changes over time. If we take this a little too rigidly, without interpreting anything, the second derivative would tell us “how the ‘change over time’ changes over time”. That isn’t very helpful. But if we use the interpretation of the first derivative as a speed or velocity, then this will shed light on the second derivative. Then the second derivative would then be a description of how the speed of an object changes over time. But we already have a word for that – acceleration! So, if we have some sort of function describing where something is, then its second derivative tells us about how that body is accelerating.

Acceleration is an incredibly important concept in physics. One of the great discoveries in the era of Galileo was that heavy objects and light objects accelerate exactly the same way when dropped from a height. Isaac Newton was able to do even better – and this is the application I’d like to talk about.

Newton’s theory of gravity can basically be reduced to two pieces. Firstly, there is his theory of force – that F = ma. This reads as “force equals mass times acceleration”. So, acceleration plays a big role. The second component describes how to calculate the force of gravity, the equation is F = G \dfrac{m_1 m_2}{r^2}. G is a constant value, m_1, m_2 are the masses of the two objects we are considering at the moment, and r is the distance between them (more specifically, their ‘centers’). When you put these together, you find that on earth, the acceleration that gravity causes is a constant number.

This is huge, because you can translate this sentence into a calculus equation. If f(t) is a function that tells you where some object I’ve thrown in the air happens to be at the moment, then Newton’s theory tells me that f^{\prime\prime}(t) is a constant number. In calculus, I can now ask myself the following question:

What sorts of functions have the property that f^{\prime\prime}(t) is a constant number?

Thus, we can see that calculus plays a key role in the theory of gravity. We actually haven’t developed enough calculus yet to solve this (we know how to take derivatives, but this question requires undoing derivatives). However, we will very soon be able to do exactly this. The rule of reversing derivatives is one of the most important parts of calculus – and it is the topic we will be addressing next in this series.

Conclusion

Not only is the first derivative of a function useful – so is its second derivative. We can use these second order derivatives to make calculations regarding gravity and in order to solve optimization problems – both of which are extremely important for engineering and all sorts of practical applications.

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