Measuring the slope of a curve

The following is a quick refresher on scalar derivatives. To compute the slope at any given point, the standard way is to typically measure the slope of the line between the point we're interested in and some secant point, which we'll call delta x:

As the distance between x and its neighbor delta x approaches 0, or as our limit approaches 0, we arrive at the slope of the curve. This is given by the following formula:

There are several different notations that you may be familiar with. One is f prime of x. The slope of a constant is 0. So, if f(x) is 9, in other words, if y is simply 9, it never changes. There is no slope. So, the slope is 0, as shown:

We can also see the power law in effect here in the second example. This will come in useful later on. If we multiply the variable by the power, and decrement the power by one, we get the following: