Derivatives

Prerequisites

You're likely to have learned about derivatives in your pre-requisite math classes.   And you may be still able to remember some useful calculus facts — such as the derivative of $x^n$ is $nx^{n-1}$ or even the chain rule: $\frac{df(g(x)}{dx} = \frac{df(g)}{dg}\frac{dg(x)}{dx}$. But for using derivatives in science (as well as any math in science), you need to have a more conceptual understanding of what the derivative means and why we need to use it. On this page, we'll discuss the general what and why issues. Detailed ways of thinking about and calculating derivatives will be given on the follow-on page. (What is a derivative, anyway?)

Why we need derivatives in science

How things change is a critical part of what science is working to understand.

When we have something ($f$) that depends on something else ($x$), we describe that by writing that $f$ is a function of $x$, $f(x)$. What we then often want to know is: when the something else ($x$) changes, how does the something ($f$) change?

This is really important in applying math to science, since three of the things we always want to do in science are:

• predict the future,
• see how turning some dial changes some result,
• figure out how important some effect is.

Predicting of the future requires knowing how something changes with time.

The dials we consider changing don't have to be real dials — they can be parameters dictating an organism's survival that can be dialed in a population by variation and natural selection.

Figuring out rates of change also tells us how important a particular parameter is. In our modeling of a physical or biological system we might say, "Oh! This change will produce this result."  But if we don't know how big the result is, we don't know whether that really is the cause of the effect we are trying to explain.

Many of the key physics concepts covered in this class such as force, pressure, or motion are connected to one another through changes. Let's look at two examples:

1. A difference in pressure between two different positions in a fluid can be responsible for the motion of that fluid — for how the blood flows in your veins, how sap moves in a tree, and how winds are created.
2. When forces are applied to an object, it may result in a change of velocity of the object with time.

For both of these examples, the basic concept is the derivative — how a change in one quantity causes a change in a quantity that depends on it. So if the pressure is a function of position, $p(x)$, the amount of flow of a viscous fluid through a pipe is proportional to how fast the pressure in a pipe is changing with distance. If the pressure drops by an amount $\Delta p$ in a distance $\Delta x$, then the flow in the pipe, $J$ will satisfy

$$\frac{\Delta p}{\Delta x} = ZJ$$

where $Z$ is a constant that depends on the properties of the fluid and the pipe.

We can define the derivative to be the slope of $p$ as a function of $x$ or how it changes over very small distances:

$\frac{dp}{dx} = \frac{\Delta p}{\Delta x}$ for $\Delta x$ smaller than we care to worry about.

We'll discuss this more below.

While you have already studied derivatives in your calculus class, using them in science might require a bit of a shift in perspective. In calculus classes, you may have focused on the mechanics of taking derivatives rather than on thinking about how to make sense of them. When we apply this math as a model of a real physical system, not all of the mathematical details are always relevant — or even correct.

The mathematics of the derivative

Notational issues

The heart of the idea of derivative using standard math notation, is that a function (a dependent variable), $f$, depends on an independent variable, $x$.  We will refer to the value that $f$ takes as $y$.  Thus we will write

$$y = f(x)$$

Be careful not to get too attached to this particular choice of letters! We will have lots of different symbols and functions.  In some situations in this class, $x$ will stand for an independent variable — something specifying a location in space.  In other situations, $x$ will stand for a dependent variable — the position of something particular, which may vary with time, and so we might have $x=f(t)$.  In other situations, $x$ will become an independent variable again, specifying which particular object we are referring to.  This can be very confusing if you think only about the symbol and not what it is meant to represent! [This argument is very similar to saying that instead of a $y$ vs $x$ graph we also study $x$ vs $t$ or $y$ vs $t$ graphs!]

We define the average rate of change in the value taken on by $f$ when $x$ changes by the expression $\frac{\Delta f}{\Delta x}$.  If $\Delta x$ is smaller than any change in the variable $x$ we care to worry about, we write the changes as $dx$ and $df$ and define the derivative as the ratio of the small changes:

$$g(x) = f '(x) = \frac{df}{dx}$$.

The notation with the prime (or dot if the independent variable is time) is due to Isaac Newton, one of the inventors of the calculus. The ratio notation is due to Gottfried Leibniz who invented calculus independently from Newton at about the same time.*  Newton's notation hides the fact that the derivative is a ratio and makes it hard to see the units of the derivative. We'll stick with Leibniz's notation.

In creating a complete and self-consistent mathematical structure for doing calculus, mathematicians often talk about "taking limits" as the change goes to zero (referring, for example, to what mathematicians mean when they change from $\Delta x$ to $dx$).  They would see our language "smaller than any change we care to worry about" as being sloppy math. Well, it is.  But we are not doing math. We are using math as a model of a system in the physical world, and the smooth ("differentiable") functions that the mathematicians like to talk about don't often exist in the physical world.

In a math book you might be shown the graph of a curve representing a function.They might then "zoom in" on a little piece of the curve showing it getting straighter and straighter as you get closer and closer in.  That's because the curve is assumed to be smooth. But in a real-world example, zooming in might make a curve look smooth and straight for a bit, but getting further in often starts to show something funny: like in the picture below.

(Source: Almquist and Melosh, Fusion of biomimetic stealth probes into lipid bilayer cores, PNAS 2010, 107:13, 5815-5820.)

In this paper, the researchers studied the properties of a lipid membrane by attaching a brush of hydrophobic molecules to the hydrophilic tip of an atomic force microscope. They then measured the force it took to push the tip through the membrane and inferred properties of the membrane.

The point is that the curve that they found on the right is not smooth. It has spikes and jiggles, and the closer you look the worse they get. Taking a derivative of this curve at this scale would be not very meaningful.  We can still smooth the curve in some way and take the derivative of the smoothed curve and get information about the rate of change of the force in the important time interval highlighted in red.

This is why we say that our derivative is the ratio of two changes when "the change is smaller than anything we care to worry about."  In the graph above, we might only be worried about what the average force is as we start to push the probe through the membrane and how much time it takes to get through.  For that, we might want to smooth the curve. If we are interested in the actual mechanism of how the individual molecules on the brush interact with the membrane, we might be interested in the details of the individual fluctuations in the graph instead of in the average derivative.

In general, in physical systems we can't take a limit to 0, since as we get too small the physics of the system changes. When we get down to the micron level, we have to worry about thermal fluctuations, and when we get down to molecular sizes we have to worry about the discreteness of matter caused by its being made up of atoms. At even smaller scales, we need quantum mechanics, and the very concept of position becomes fuzzy.

So if your math course fussed about limits, it's fine to learn about it for the sake of the math —  limits allow the math of derivatives to be proven and introduced very nicely. However, when thinking about using derivatives to model something in a physical system, it's much better to think about them as ratios of small changes.

This view also clarifies something that confuses many students. Although we talk about a derivative "at a point" — so we write "$g(x) = f'(x)$" — the derivative of $f$ at the point $x$ — thinking in terms of the ratio of changes shows us that a derivative is really about the values of $f$ at two different points. See the discussion of velocity for an example.

An interesting book on the battle for credit that went on between Newton and Leibniz is Jason Bardi's book, The Calculus Wars: Newton, Leibniz, and the Greatest Mathematical Clash of All Time.

Julia Gouvea, Kerstin Nordstrom, and Joe Redish 9/9/13

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