Calculus! #26: Early Transcendentals 2.7A

Derivatives and Rates of Change

Holy balls, we did it. The glorious derivative. Not only does this mean you’re about to get to some real live Calculus, it means after this section is complete I can go back to also posting on physics stuff!

I was trying to think about the smart way to explain derivatives, and here’s what I’ve decided: Since you’re primed from all the limits stuff, I want to continue with that. But, at the end of this section, I’ll do a little bit about intuitively grokking derivatives. If you don’t have a sense of what you’re actually doing you will get screwed hard later on.

But first, let’s get into the math.

So, imagine you have a function in 2D. This function intakes x values and spits out f(x) values (aka, y values). Say you mark of two points on this function. That’ll give you two sets of (x,y) values. And, if you remember from pre-calc, that’ll give you the slope of a line between those two points. For now, let’s call those points P and Q, and say Q is (x, f(x)) and P is (a, f(a)).

Here’s how that looks mathematically, assuming m is the slope, and P and Q are the points:

m_{PQ} = \dfrac{f(x) - f(a)}{x - a}

Try to think of this visually. Imagine those two points as defining a slope that forms a hypotenuse, and draw an imaginary triangle using it. Now, imagine we take the two points and bring them closer and closer together. This changes that triangle, such that the hypotenuse shrinks and shrinks and shrinks.

You’ll note that as the distance between the two x values (we’ll call them x and a) shrinks, the hypotenuse does too. You also notice that the hypotenuse becomes “tangent” to the curve as you get smaller and smaller. That is, it just barely touches the curve, and at that one point where it touches, the curve and the hypotenuse look parallel.

It doesn’t matter where you start the two points on the curve. If you keep bringing them closer together and drawing the slope all the while, eventually you’ll be tangent to the curve at some point.

Now, in the past, you wouldn’t know how to proceed here. You might be able to approximate the tangent at a particular point by taking a smaller and smaller distance between x and a. BUT NOW YOU’RE MORE POWERFUL! Thanks to your knowledge of limits, you can see what happens when you take that distance infinitely small.

m = \lim_{x \to a} \dfrac{f(x) - f(a)}{x - a}

That is, the slope of the tangent at (a, f(a)) is equal to the limit of the slope as you take x and a really close to each other.

Got it? Well, then you’ve basically got the derivative. The book gives it some slightly prettier mathematical language by introducing h.

This is essentially the same equation, made nicer by defining h as the distance between the two points. So, rather than deal with two points, we’re dealing with a point and a distance. It’s not really different, but it results in this equation that you should etch  into your brain:

m = \lim_{h \to 0}\dfrac{f(a+h) - f(a)}{h}

That is, the slope of the tangent at a point can by had by taking the function at a and at (a + h), finding the slope between those two values, and then taking the limit as h gets closer to 0. By closing the gap between a and a point beyond, you find the slope of the tangent line at a.

This is the definition of a derivative. The book segues into talking about velocity here, but I want to pause for a minute to make sure you get derivatives.

When I was first learning these, I learned HOW to use them, but didn’t have a good grasp of what they are. Yes, you’re calculating the tangent to a curve, and yes you’re taking a limit as two points get closer and closer. But, you’re also saying something more general.

Remember what a function is: It’s a way to get from one set of values to another. For example, the function that defines all odd numbers could be stated like this: Over here, you’ve got all the integers (given by I). Over there you’ve got all the odd integers (given by O). To get from here to there, multiply each I by 2, then add 1.

The derivative is something you can do to that function to ask the data another question. The question you’re asking is “at a given point, how is one set of data changing with the other.” More commonly, you might say “how is x changing with y?” So, for example, in an equation like f(x) = x^2 f(x) grows much faster than x. So, when you move between your data sets, you’ll find when x goes from 1 to 2, the difference between f(1) and f(2) is only 3. When you go from f(10) to f(11), the difference is 21.

As you can imagine, things get even trickier if you start including things like logs and trig functions, and fancier stuff. The derivative allows  us to ask the data how it’s changing, and this is a powerful thing indeed.

Hopefully that serves to give you a little better understanding of what you’re dealing with. I don’t want to belabor it too much, but my sense is that a lot of students learn the derivative as a tool, rather than a concept. We’ll get to the tool part, but it’s important that you take some time now to understand what a derivative actually is.

Next section: Velocities

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