2.2 The Definition of the Derivative

The term derivative comes up a lot in math, science, engineering, economics, and finance. It is a powerful tool in the real world, yet it has profound philosophical implications as well, answering questions that go back as far as ancient Greece. In this section, we will combine the elements from section 2.1 with the notion of limits which we discussed in Chapter 1 to build the derivative.

For this section, we are going to consider two points on a curve:

$(x_0, f(x_0)) \ , \ (x_0 + \Delta x, f(x_0 + \Delta x))$

The symbol $\Delta x$ (read “delta x”) just stands for the horizontal distance separating the two points. The equation for the secant line connecting the two points is:

$y - f(x_0) = m (x - x_0)$

where the slope $m$ is given by:

$m=\frac{f(x_0+\Delta x) - f(x_0)}{x_0+\Delta x - x_0} = \frac{f(x_0+\Delta x) - f(x_0)}{\Delta x}$

This expression for the slope of the line is going to be the main subject of this section. Let’s put together everything we have so far into an animation. For this demonstration, we’ve chosen the function $f(x)=x^2 - x - 2$ . Below, you can move the value of $x_0$ around with the slider. The value of $\Delta x$ (denoted as $\Delta_x$ in the animation) is fixed to 1 for now.

Notice that as you change $x_0$ , the slope changes too. Now, let’s combine this with the idea of a limit. As we shrink the value of $\Delta x$ , the secant line at $x_0$ approaches the tangent line! You can explore this in the next animation, which allows you to change both $x_0$ and $\Delta x$ (again, written as $\Delta_x$ in the animation).

This is exactly the notion of the derivative. The derivative of a function $f$ at a point $x$ is the slope of the secant line in the limit that $\Delta x \rightarrow 0$ . Let’s formalize this with a definition:

$\frac{d}{dx} f(x) = \frac{df}{dx} \equiv \lim\limits_{x \to 0}\frac{f(x+\Delta x) - f(x)}{\Delta x}$

The expression $\frac{df}{dx}$ (read “d-f-d-x”) is called the derivative of $f$ with respect to x. Think of it as an instantaneous rate of change; it is the ratio of the infinitesimal change in $f(x)$ to the infinitesimal change in $x$ . Notice that this is a new function of $x$ . We should expect this because the slope of the tangent line changes as we change $x_0$ in the animation above.

You might ask if we are even allowed to do this. After all, what does it mean to take the ratio of two things that are infinitely small? Even more problematic is the fact that if we plug in 0 directly into the definition of the derivative, we get $0/0$ . This calls into question whether or not the limit exists at all! We will need to check this carefully every time we encounter a new function. Luckily, we will find that certain rules apply generally to many functions of the same class. (For example, we will soon find a single rule to calculate the derivative of any polynomial function.)

Let’s start with the example from the animation. Our function is $f(x) = x^2 - x - 2$ . Let’s calculate the derivative using the definition:

$\begin{aligned} \frac{df}{dx} &= \lim\limits_{x \to 0}\frac{f(x+\Delta x) - f(x)}{\Delta x }\\ \frac{df}{dx} &= \lim\limits_{x \to 0}\frac{\left((x+\Delta x)^2 - (x+\Delta x) - 2\right) - \left(x^2 - x - 2\right)}{\Delta x}\\ \frac{df}{dx} &= \lim\limits_{x \to 0}\frac{(x^2+2x(\Delta x) + (\Delta x)^2 - x - \Delta x - 2 - x^2 + x + 2}{\Delta x}\\ \frac{df}{dx} &= \lim\limits_{x \to 0}\frac{\cancel{x^2}+2x(\Delta x) + (\Delta x)^2 \cancel{- x} - \Delta x + \cancel{- 2} \cancel{- x^2} \cancel{+ x} \cancel{+ 2}}{\Delta x}\\ \frac{df}{dx} &= \lim\limits_{x \to 0}\frac{2x(\Delta x) + (\Delta x)^2 - \Delta x}{\Delta x}\\ \frac{df}{dx} &= \lim\limits_{x \to 0}2x + \Delta x - 1\\ \frac{df}{dx} &= 2x + 0 - 1\\ \frac{df}{dx} &= 2x - 1\\ \end{aligned}$

After cancelling out all the terms that we can in the numerator, every term that remains has at least one power of $\Delta x$ . This allows us to cancel the $\Delta x$ in the denominator, solving the problem of dividing by zero. The left over factor of $\Delta x$ can then be set to 0 when we take the limit. It is very important to fully reduce the fraction and eliminate the $\Delta x$ in the denominator before you evaluate the limit. This will not always be straightforward, as we will see later. The moral of the story is to be very careful when handling this limit.