So, now that you get the basic idea of a derivative, let’s apply it to something mathematicians hate… real life.
Say you have a function (we’ll call it s), which is a description of position over time. Mathematically, that’s f(t) = s, where t is time.
This function is handy enough. You can ask it your position at any given time. But, what if you want to know the velocity at a given time (aka, instantaneous velocity)? Well, remember that velocity is just change in position over time. You have both of those elements – position and time – already in your function. So, to find velocity we just need to figure out how they change with each other at some point.
This sounds like a job for derivative!
To make it interesting, let’s describe a real situation. It could be any situation really… oh! Let’s say I’m doing your mom.
My wiener’s position is described by f(t) = s. Nevermind what the actual function might be. For now, let’s just say that position is a function of time. Now, say I promised yo mama I’d go easy on her, and you attached a device to my pelvis that measures its position with respect to your mom’s hot ass, to make sure I’m treating her right.
You check it at 11pm, and I’m 6 cm away. You check 2 minutes later and it’s 5 cm. You check it 2 minutes later still and it’s at 0.1 cm. You check 2 minutes later and find it’s at 3cm. From the small set of numbers, you can’t deduce much. You know that position is changing, but you’d like to be able to check velocity at a particular point. That is, you’d like to know how position changes with respect to time.
So, you start checking at smaller and smaller time intervals. Then, it occurs you that you could do a limit equation here.
That is, the velocity of my wiener at a given time equals the limit of change in position over change in time as change in time goes to 0.
You manually take the intervals closer and closer to 0 at many points. At each point, you find that I’m loving your mom, slowly and sensually. This is much more horrifying than the rough sex you’d expected to see.
Of course, you can’t manually check every point. OR CAN YOU? We’ll get to that in chapter 3 when we get more into the tool set of derivative solving techniques. For now, we’re going to do some more basics.
You already have the mathematical definition of a derivative. Let’s get into the symbols. In my experience, there are 3 common ways to denote derivatives: tics, dots, and Leibniz.
Tics are quite common, probably because they’re easy to do legibly. They go like this:
f(x) is a function
f’(x) is its derivative
You can also take derivatives of f’(x). Each time you do so, just add another tic:
f”(x) is the second derivative
f”’(x) is the third, and so on.
In my experience, the verbal expression of this is to just say “prime” for each tic. So, the first derivative would be pronounced “f prime,” the second would be “f prime prime,” the third would be “f prime prime prime” and so on.
The next technique is to put little dots above your function. I don’t use this much, but I know some people who do. It works like this:
is the first derivative, pronounced “f(x) dot.”
is the second derivative, pronounced “f(x) dot dot” or “f(x) double dot.”
And so on.
The Leibniz notation expresses derivatives as a fraction, but the book leaves that for the next section, so I will to.
Last for this part, we get two definitions. One is yet another phrasing of the definition of a derivative. Here it is:
By now, it should be pretty clear what that means.
Lastly, this subsection closes with a restatement of what we’ve learned so far, regarding the tangent line: “The tangent line to y = f(x) at (a, f(a)) is the line through (a, f(a)) whose slope is equal to f‘(a), the derivative of f at a.
In other words, if you take the derivative at a point, you get the slope at that point. Wooh!
Rates of Change
This section is worth reading, but I won’t linger on it. Here’s the basic idea:
Remember how the derivative of position with respect to time is velocity? Well, using the derivative, we can figure out how lots of things change with respect to other things. For example, you can measure change in population with respect to available food. You can measure change in volume with respect to time, of an expanding balloon. You can measure anger with respect to stupidness of workplace policies. These are all derivatives. Just take the limit as the thing following “with respect to” shrinks in interval.
So, in the case of how anger changes with respect to workplace stupidity, you’d take the derivative and get a slope for a given point. The important thing here, that a lot of people get hung up on, is that neither factor relates to time. Because time is so common, students start thinking that “take the derivative” is synonymous with “take the derivative with respect to time.” This is commonly the case, but it is not always the case. The derivative is how one thing changes with another thing. It is not necessarily how one thing changes with time.
The rates of change section basically deals with this expanded idea of a derivative. It’s worth going over before you start in on the next problem set.