Second and higher derivatives.

Think of a function. Any function. (Well, make it one that you can differentiate). Differentiate it. What is the result?

Whatever function you chose, there's something I can say about the result: it's another function, isn't it?

(OK, it's not a great party trick, but at least it's true).
This new function gives you the slope of your original function, which is another way of saying that this new function tells you the rate at which your original function changes as you change the variable it depends upon.

For example, if you chose sin(X), the derivative is cos(X), and so the slope (rate of change) of sin(X) at any value of X is the value of cos(X) at that point.

This was all covered in earlier sections. The new point is this, if we've just ended up with another function as a result of our differentiation, then we could differentiate again, i.e.differentiate this new function.

Returning to the sin(X) example, our first differentiation gave us cos(X), so if we differentiate again we'll get -sin(X).

Of course we could do this forever, keeping on differentiating over and over again. But what do these new derivatives represent, in the way that the first derivative gives us the slope? And how do we write these new derivatives, in the way that we use dY/dX for the first derivative?

The second question is easier to answer. If we start with a function Y(X), then differentiating once gives us dY/dX. Then differentiating again gives us the so-called second derivative of Y(X), written as d²Y/dX².

Notice where the "2"'s are placed in this expression, it's not the same thing at all as (dY/dX)², which is the first derivative multiplied by itself.

Going on, if we differentiate again, we'll get d³Y/dX³. This would be called the third derivative, not surprisingly. And so on. If we differentiate n times, where n is any positive integer, then we get the so-called n^th derivative, dⁿY/dXⁿ.

All very straightforward.
Now what do these so-called "higher" derivatives represent? Each new derivative is the rate at which the previous derivative changes as X changes.

We'll just think what this means for the second derivative. Since the first derivative is the slope of the function Y(X), the second derivative d²Y/dX² tells us how that slope is changing, whether its changing quickly or slowly as we move on in X.

To give some idea of what this means, let's have another look at the graph of sin(X). The slope is given by cos(X) and this mysterious second derivative is -sin(X) as we worked out just now. That means that the second derivative is largest where sin(X) itself is largest, at the top and bottom of its oscillation.

Look at the slope near these top and bottom regions, and in particular look how quickly the slope is changing there.

In contrast, in the regions of the curve near to the horizontal axis the slope hardly changes at all. For instance look at the portion of the curve between X=2 and X=4. The slope looks like about -1 for that whole portion. But then around X=4.5 the slope changes very quickly, from about -1, through zero, to about +1, within a very short X-distance.

So the second derivative tells us how rapidly the slope is changing, in the same way that the first derivative tells us how rapidly the function itself is growing or decreasing.

One further point: the more common word for the rate of change of slope is the "curvature". A large value for the second derivative means high curvature. The function sin(X) has its highest curvature at the top and bottom of its oscillation.