The Derivative, Part 1

In my last post, I showed that the rate of change of any function that plots as a straight line (a linear function) has a constant rate of change and that value is the gradient of the line. However, for a nonlinear function, its rate of change depends on the value of x, that is, where you are on its graph. I also said that a function’s rate of change is called the derivative of the function and that is what I will call it from now on.

Graphically, the derivative at a particular x value is the gradient of the tangent line at that point:

We would like to find an easy way to mathematically find this value as opposed to graphing the function and estimating the tangent line’s gradient at the desired points. Clearly, as seen above, the derivative is another function of x as its value changes depending on what x is. There are several ways to denote the derivative, but we will start with f’(x) (read as “f prime of x”). We would like to find what f’(x) is given a function f(x).

I know that the following derivation of the derivative may look complex and begs the question about how easy it will be to find the derivative, but following this will help solidify your understanding of what a derivative is and the final result will be used many times to find the easy results for various function forms.

We begin by taking the graph of a function and drawing a secant line (a line that connects two points on the graph) and calculate the gradient of that line:

We want to know the gradient of the estimated tangent line which we are using to approximate the tangent line at x. From your study of linear equations, you know that the gradient of a line can be found from any two points on the line. The two points on our estimated tangent line are (x, f(x)) and (x+h, f(x+h)) where h is a small distance away from x. Using these two points, we find the gradient by calculating the rise from the first point to the second point divided by the run between the two points. The rise is the difference between the y coordinates and the run is the difference between the x coordinates (h):

\[\text{gradient} \ =\ \frac{f( x+h) -f( x)}{h}\]

Now what happens as h gets smaller? The estimate should get closer to the actual value we are seeking. The below graphic from IkamusumeFan [CC BY-SA (] illustrates this:

So it appears that we are interested in what our estimated gradient approaches as h approaches zero. This is, in fact, the formal definition of a function’s derivative. Remember my post on limits? Using limit notation then, the definition of the derivative is

\[f'( x) \ =\lim _{h\rightarrow 0} \ \frac{f( x+h) -f( x)}{h}\]

Notice that if we just substitute zero for h to evaluate the limit, we get the indeterminate form 0/0 as explained in my prior post. So again you may be saying “this doesn’t make finding a derivative easy at all”. At this point, you are correct. But in my next post, I will show how this definition is used to simplify derivatives.