FIRST, THE BACKSTORY...
The Ancient Greeks were excellent at geometry, and even got pretty good at dealing with circles… but all their curve and circle work pertained to static models. A circle drawn on a sheet of (whatever they used for) paper, or a curve sketched in the sand.
It was not until the great Galileo that math really become and stayed ACTIVE.
Galileo studied motion. And he made a great break-through with this notion that– though a moving object only cuts one path through the world– we can think of its motion of being comprised of TWO directions at once. For example, consider the projectile… A projectile can be said to have two components of velocity… 1) its vertical motion, and 2) its lateral motion.
We can think of these two directional components as two arrow-headed lines, with their lengths representing how far the object travels in that arrow’s direction. From the point of origin, we can draw one of the arrows (doesn’t matter which one we start with) and then, starting from end of that arrow, draw the other arrow, so that where the end of the first arrow meets the beginning of the second, a right angle is formed.
If we then draw a line from the origin of the first arrow to the end of the second, that line will form the hypoteneuse of the right triangle formed by it and the two arrows. This hypoteneuse represents the approximate path (both direction and distance) taken by the moving object. This approximation of the path is known as the resultant.
We can obtain more and more realistic resultants as we examine the path in smaller and smaller increments. For example, you can draw a series of smaller arrow-pairs showing how far the object went vertically and horizontally after one second, another pair representing the 2nd second, another for the 3rd, and so on… If the velocity is not constant in both directions, then the series of resultants we draw connecting the arrows will look like a kinked curve.
Each resultant gives us the ratio between the change-in-height and the change-in-distance of the projectile during the time increment. This ratio is central to the mathematics of movement and has several names…
The Resultant = the Slope = the Rise-Over-The-Run = Change-In-y per Change-In-x = the Differential = the Tangent.
I like to call this Hammering Shield’s Long Equation Of Same-Same.
Let’s take a sec to explore the tangent aspect... As these right-angled pairs of component-directional arrows cover smaller and smaller intervals, the graph for an object moving with a steadily increasing or decreasing velocity will start to look more and more like a curve. If an increment grows small enough, its resultant will skim the curve at one and only point. In other words, the resultant will be tangent to the curve.
Sometimes, the symbol “delta” stands for “change-in…”, so that, for instance, change-in-y divided by change-in-x becomes: “delta-y” over “delta-x.”
For a velocity equation, the “delta-y” value typically represents “change-in-distance” and “delta-x” is “change-in-time”. The tangent in this case represents… 6) the Velocity. I don’t put this in my Same-Same Equation because it is only true if the axes are distance and time.
In Calculus, when we “take the Derivative” of a position-equation, we are simply finding the Differential (aka Velocity) as delta-x becomes so small that it approaches zero, meaning that the change in time becomes infinitesimally small. The velocity as x approaches zero is called the “Instantaneous Velocity.”
Boyer warns us that the term ‘instantaneous velocity” makes it look like that the time interval is to be regarded as zero. If this were truly the case then the ratio change-in-distance over change-in-time “predicates the very case which mathematics is compelled to exclude because of the impossibility of division by zero.”
Instead, Boyer suggests that we do not think of this Differential, or “DIfference Quotient” (delta-y / delta-x) as a true ratio. It cannot really be the change-in-position over the change-in-time– for any velocity which is purportedly “instantaneous” HAS no change in time (or distance). Boyer advizes us to keep in mind that… “this is not a velocity in the ordinary sense and has no counterpart in the world of nature, in which there can be no motion without a change of position.”
Let’s take an example:
Let us say that y = x^2 + 4x + 4. This equation is called a FUNCTION, because the value of “y” can be seen to depend on the value of “x”– in other words, to be a “function” of x (and yes, philosophically speaking, you could also make the case that the value of x depends on the value of y, but that won’t really help us right now). Since y is a function of x, we can replace the “y” with “f(x)”, meaning “function of x.”
The Differential of this function tells us HOW MUCH y or f(x) is changing PER change in x. Mathematically, we would write the Differential like this…
[ f(x+ delta-x) – f(x) ] / [ (delta-x + x) – x ]
That simply means: the difference between the second and the first y-values produced… PER the difference between the second and first x-values. In other words, how much the y-value changes when the x-value changes.
DERIVING THE DERIVATIVE:
To take it to the next level, we can find the Derivative of the function by allowing delta-x to approach zero. Thus, the Derivative is the Limit As x Approaches Zero of the Differential.
Now… this is the tricky part. This is why Newton and Leibniz get to sit at the adult table of history… if you do this the wrong way, you’ll just get zeroes. Not helpful. Here’s how Isaac and Gottfried did it…
The denominator of the Differential ends up being…
(delta-x + x) – x …allowing “dx” to stand for “delta-x”, we can re-write this as…
= (dx + x) – x
For our equation, f(x) = x^2 + 4x + 4, the numerator of the differential would be…
[f(x+ delta-x)] – f(x)
= [(x + dx)^2 + 4(x + dx) + 4] – [x^2 +4x + 4]
= x^2 + 2xdx + dx^2 + 4x + 4dx + 4 – x^2 – 4x – 4
= 2xdx + dx^2 + 4dx
when we divide this by the denominator of “dx“, we get…
2x + dx + 4
Now, if we allow dx to approach zero, the value will become so infinitesimally small that even strict mathematicians (nowadays) allow us to drop dx from the equation, so that we are left with…
2x + 4
Thus, 2x + 4 is the Derivative of the function x^2 + 4x + 4. You Calc-students knew the shortcut way to “take the Derivative,” but now you know what’s going on behind the trick.
And bonus… the Derivative is equal to all those elements of Hammering Shield’s Long Equation Of Same-Same anywhere on the curve! This makes the Derivative a very useful tool, indeed… Want to know what the Tangent is where x=3? Use the Derivative. Want to know what the Slope is where x=105? Use the derivative.
Boy, are you lucky you ran into me!