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Differentiation: Calculus for the Hard of Thinking

June 15th, 2009


Most of us tend to frown and worry a little when we hear the word calculus. We prefer not to think about it too much. “Calculus” generally has an aura of “proper” mathematics about it, some hint of complexity and difficulty, but few of us actually know what it is or what it means. It sounds vaguely like “calculation”, so it must be hard, right?

Well, sort of. It does get hard. But it’s really not that difficult to begin with. The word itself is Latin, and refers to the type of pebble you might use for counting. This is an excellent place to start because that’s just what calculus is: breaking things down into smaller and smaller chunks (nowadays it’s known as tending towards a limit, but we don’t need to go into that now). It is used for working out gradients — rates of change — and calculating areas of fiddly objects (ones with nasty, curved surfaces). All you need now is a little knowledge of gradients and a little knowledge of indices.

When we start the study of calculus, we begin with differentiation. Differentiation is the process of looking at a graph and working out how y changes with respect to x: in other words, how steep the curve — or line — is. Every curve or line on a graph has its own equation, which is its unique description — its DNA, if you like. A line might be y= 2x + 3. Each time you have an x value you plug it into the equation to find the corresponding y-value. In fact the equation contains the following bits of information: the gradient (2x) and the point at which the line crosses the y-axis (+3, because x=0 at this point). A curve might have equation y= 2x² + 2x + 3. Now a line is simple. It is straight and its gradient is constant. You can see it in the equation. A curve is always changing. Every point on it has a different gradient (though the same value might recur at different points).

So you need to do some work to find the gradients. What you will need to do is find a general formula for the gradient of that curve. When we plug any given x-value into this formula, this will give us the gradient at this point.

It works like this. Take a curve. Choose two points. Call the first one P. We will try to find the gradient of the curve at this particular point. So we will be using this one all the way through. Draw a line between P and any other point. Calculate the gradient of this line in the normal way: y2-y1/x2-x1. This gives you a really rough approximation because though the line touches P, it goes a long way out from it on its way to the other point. Now draw a second line, from the original first point P to a new second point closer to P. Do this again, and again and again, making the distance between P and the new point smaller each time. You will find that whatever your initial value for the gradient of the line (say, 4.666666), the subsequent values, as you approach your first point, tend towards a certain number. They might go like this: 4.66666, 4.5555, 4.35555, 4.22222, 4.1.

You cannot work out a gradient from just one point because you cannot draw a line between P and P, but you are getting closer and closer to this one point, which is an increasingly good approximation of the gradient of the curve at your first point. Less and less change from P is happening as you get closer to it.

We say, therefore, that the gradient at point P, tends to 4 as the difference between x and y tends to 0.

What you have done is make the difference between x and y smaller and smaller to get a better and better view of the gradient at x.

You have differentiated.

Clever, isn’t it? It gets better, because if you do this algebraically, which is awkward but not impossible, you can see a simple rule emerging for the calculation of gradients. We’ll skip the algebra for now and move onto the rule.

The effect of it is that when you have your curve: y= 2x² + 2x + 3, there is one simple rule for differentiating: multiply the power of x by its coefficient, then bring that power down by one.

Eh?

Let’s do it in bits. 2x². 2×2 = 4 (multiplying the power by the coefficient — the number in front — of x). 2-1 = 1 (bringing the power down by one). 2x² becomes 4×1 or just 4x because any value x raised to the power of 1 is just x.

2x, the next bit, already means 2×1. 1×2 = 2, 1-1 = 0. So, 2×1 becomes 2×0. X, (that is, anything at all) raised to the power of 0 is 1. So we just have 2.

The last bit is 3. There is no x here at all, just 3. Now this means that we have 3×0 or 3×1, in other words just 3. Now trying to multiply 0, the power of x here, by 3, simply gives us 0. So we lose the three. In fact, any number without an x-value, when differentiated, is lost.

So y= 2x² + 2x + 3, which is a curve, differentiates to give dy/dx (the differentiation symbol) = 4x + 2. This is our general formula for the gradient of this curve, and whatever x-value we plug into it, will give us the gradient of the curve at that x-point. Remember, the gradient of a curve is always changing, so this will be different for each x-value.

Not difficult, not nasty, and not impossible. Yes, this is a very basic account of a wide-ranging aspect of mathematics, but it is a good place to start. So if you are afraid of calculus, go ahead and try it! You will be amazed with your new found powers of differentiation!


This article on calculus and differentiation was written by Jack Blair, a relatively new Constant Content author who has produced three articles and two sales.


One Response to “Differentiation: Calculus for the Hard of Thinking”

  1. comment number 1 by: Ireland5

    Thanks - I needed this. Interesting article! Love the title…

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