Video Lecture

In this video we aim to adhere to the number one mathematician’s rule: Embrace Laziness!

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Up until now, we have computed derivatives at specific (concrete) $x$-values via the difference quotient, but this can be a time consuming, not to mention tedious, process. Often we need to know rates of change at several points for a given function, or you may be looking for when the derivative has a specific property (like when the derivative is zero). Doing this one point at a time would be, at best, incredibly annoying... at worst (and almost surely in many cases) it isn’t even possible. Luckily, there is a better way!

One of the mathematician’s credos is to solve problems in as much generality and abstraction as you can generally manage. This may seem like an unnecessary and confusing practice - after all, why solve for all the possible configurations of a problem, when you are only presented with a specific example to solve originally? But this kind of thinking is actually embraced for a practical reason (and also because mathematicians are a quirky bunch, but mostly the practical thing).

One the one hand, solving for things in general for a mathematician is a way to get at the underlying structure and the heart of “why” something is true, or “how it works” at a truly fundamental level. This is often a key component to the “beauty” that mathematicians describe as finding in their work. Being able to boil something incredibly complex and detailed down to only one or two key concepts and still capture the entire essence of the problem they started with is one of the true sources of joy for a research mathematician.

However, as I’m sure you are saying by now; you are not a research mathematician, and you probably don’t agree (or at least don’t especially care) about what mathematicians think is beautiful in mathematics. Nonetheless, I would argue there is still a good reason for you to embrace this way of thinking - one born of pure self interest and nothing so abstract or ephemeral as beauty or joy.

I know... bold claim right? But let me make my case before you roll your eyes so hard you detach a retina...

Using the correct level of abstraction results in slightly more work up front, and massively less work later on. It allows you to abide by the old adage; “Work smart, not hard.” Put another way, this brings us back to our original mantra - embrace laziness!

Instead of solving derivatives at specific points, we want to start solving the limit in terms of a general point, $x$. To do this, we will need to do marginally more work initially (although, fair warning, even though it isn’t a lot more effort, it can look more intimidating, at least initially) but the payoff will be worth it. Consider, for example, the function $f(x) = x^2 + 3x - 1$. If we wanted to determine the derivative of this function at $x=1$, $x=-1$ and $x=0$ we could either;

(a)

Solve each of: $\displaystyle \lim \limits _{x\rightarrow -1} \frac {f(x)-f(-1)}{x-(-1)}$, $\displaystyle \lim \limits _{x\rightarrow 0} \frac {f(x)-f(0)}{x-(0)}$, and $\displaystyle \lim \limits _{x\rightarrow 1} \frac {f(x)-f(1)}{x-(1)}$.
Or

(b)

Solve $\displaystyle \lim \limits _{\Delta x\rightarrow 0} \frac {f(x+\Delta x)-f(x)}{(x+\Delta x)-x}$.

Let’s start by solving the initial three cases - note that this is presented mostly for comparison against the following second version, so you needn’t worry about following it super closely.

First we wanted the derivative at $x=1$.

Now we want the derivative at $x=0$.

And finally, the derivative at $x=-1$.

So we have done the computations 3 times and gotten that the derivative at $x=-1$ is $1$, at $x=0$ it is $3$, and at $x=1$ it is $5$. Now, let’s see how our second version works:

Ok, so the calculation was pretty involved, but not exactly difficult so much as there was just lots of stuff we needed to track carefully to avoid errors. But notice that the length of the calculation is about on par with any one of the previous calculations (indeed, if you look closely, the steps are actually very similar as well) and now, using this second approach, we have reduced the limit all the way down to $2x+3$. Importantly, we have actually technically “evaluated the limit” (meaning that we no longer have the “$\lim$” in front), so we don’t even have to worry about calculating a limit anymore.

So what does this mean? This function represents the derivative of $f(x)$ at the point $x$, for any input $x$! In other words, if we want to know what the derivative is at $-1$, $0$, and $1$, we can simply plug in those values as the $x$ value into $2x+3$ to get $1$, $3$, and $5$ respectively; a quick and easy computation. So instead of doing a slightly easier difference quotient several times like we did above, we do a slightly harder one once to get a function that represents the derivative at any point!

Noticing that we can now plug in any value we need the derivative for into this function shows us why it’s more efficient to solve this more abstract version... but that’s not even the most useful part! What if we want to know something like, “where is the derivative equal to zero?” (a question we will ask so many times in the application section that it will begin to haunt your dreams!) It isn’t really clear how we could even begin to figure that out if we are using concrete values in the difference quotient. But now that we have a function that represents the derivative at any point $x$, we can simply set this function equal to zero and solve for $x$.

Nice and easy!

This way of solving for a function that represents the derivative of $f(x)$ is so common and so useful that we even have a conventional notation! When we solve for this derivative function of some function $f(x)$ we typically denote it as $f'(x)$ (pronounced “$f$-prime of $x$”). Similarly if we find a derivative function of functions denoted by other letters, say $g(x)$ or $h(x)$ or $r(x)$, we would denote the derivative function by $g'(x)$, $h'(x)$, and $r'(x)$ respectively. There are other notations associated with taking derivatives in this abstract sense, but we will cover those as they come up.

So we have now seen that it is often slightly harder to solve the difference quotient limit in a general way (to get the derivative function), but the payoff is huge; specifically it is a function - which means it represents the derivative at any point. This allows us to solve the derivative for lots of inputs easily, without having to redo the difference quotient computation, and it also lets us solve for which inputs yield a desired derivative value (like when the derivative equals zero), which is not at all clear when just looking at the difference quotient. Finally, this kind of thing is so common and useful, that it has it’s own conventional notation, sometimes referred to as “prime” notation; for example, the derivative function of $f(x)$ is written $f'(x)$.