To begin developing a knowledge of the derivatives of common functions, and the rules we use to find them, let’s start with the most basic of functions: I. Constant; II. Linear; and III. Power Functions. Let’s examine each in turn.

I. Constant Functions

We consider first constant functions: $f(x) = c,$ where $c$ is a constant.

As Exploration 1 demonstrates,
\[ \bbox[10px,border:2px solid blue]{
\begin{align*}
\textbf{Constant Functions:} & \\[4px] \text{If $f(x) = c,$ where } c &\text{ is a constant, then $f'(x) = 0.$}
\end{align*} }\] Here are three ways to make sense of this result:

II. Linear Functions

Let’s consider next linear functions, which we’ll write in the familiar form $f(x) = mx + b.$

As Exploration 2 demonstrates,
\[ \bbox[10px,border:2px solid blue]{
\begin{align*}
\textbf{Linear Functions:} & \\[4px] \text{If }f(x) = mx+b, &\text{ then $f'(x) = \text{slope} =m.$}
\end{align*}
}\] Here are three ways to make sense of this result:

III. Power Functions

Let’s consider now functions of the form $f(x) = x^n,$ such as $x^2,$ $x^3,$ $x^{1/2} = \sqrt{x},$ and $x^{-1} = 1/x.$ We’ve actually computed each of those particular derivative functions earlier. [We’ll add these links once we make Chapter 3 available.] We’ve provided those results below, in both graphical and equation form.

Do you notice a pattern in how the derivative functions are related to the original functions? Look at this table for more examples, and the last entry, which shows how to summarize the pattern:
\[ \begin{array}{c|c|c|c|c|c|c|c|c}
f(x): & x^{-3} & x^{-2} & x^{-1} & x^0 & x^1 & x^2 & x^3 & x^n\\
\Downarrow & \Downarrow & \Downarrow & \Downarrow & \Downarrow &\Downarrow & \Downarrow & \Downarrow & \Downarrow\\
f'(x): & -3x^{-4} & -2 x^{-3} & -x^{-2} & 0 & 1 \left(x^0 \right) & 2x^{(1)} & 3x^2 & nx^{n-1}
\end{array} \] Of course the fact that we can recognize a pattern from a small set of examples is not sufficient for us to conclude that the pattern always holds; instead, we must prove that we have in fact developed a rule we can trust, or determine when the rule holds and when it doesn’t.

Using the tools we already have, we can in fact prove that the pattern holds for any positive integer value of n:

Furthermore, later we’ll also be able to show [Link to be added] that the rule also holds for fractional values of n, such as
\[ \begin{array}{c|c|c|c|c|c|c|c|c|c|}
f(x): & x^{-2/3} & x^{-1/2} & x^{1/2} & x^{2/3} & x^n\\
\Downarrow & \Downarrow & \Downarrow & \Downarrow & \Downarrow & \Downarrow\\
f'(x): & \frac{-2}{3}x^{-5/3} & \frac{-1}{2} x^{-3/2} &\frac{1}{2}x^{-1/2} & \frac{2}{3}x^{-1/3} & nx^{n-1}
\end{array} \]

But wait, there’s more [Link again to be added]: the rule also holds for decimal values of n that can be expressed as a fraction (e.g., $0.4 = \frac{4}{10}$) – meaning all rational values of n – such as:
\[ \begin{array}{c|c|c|c|c|c|c|c|c|c|}
f(x): & x^{-0.4} & x^{-0.2} & x^{0.2} & x^{0.4} & x^n\\
\Downarrow & \Downarrow & \Downarrow & \Downarrow & \Downarrow & \Downarrow \\
f'(x): & -0.4x^{-1.4} & -0.2x^{-1.2} & 0.2x^{-0.8} & 0.4x^{-0.6} & nx^{n-1}
\end{array} \] And finally, the rule also holds [Final link to be added] for non-rational values of n (values that cannot be expressed as a fraction) such as $\pi,$ e, and $\sqrt{2}$:
\begin{array}{c|c|c|c|c|c|c|c|c|c|}
f(x): & x^{\pi} & x^e & x^\sqrt{2} & x^n\\
\Downarrow & \Downarrow & \Downarrow & \Downarrow & \Downarrow \\
f'(x): & \pi x^{\pi-1} & ex^{e-1} & \sqrt{2}x^{\sqrt{2} – 1} & nx^{n-1}
\end{array}

Hence, the rule holds for all real values of n. Although we haven’t proven it fully yet, from this point onward we will use the general rule, known as the Power Rule:
\[ \bbox[10px,border:2px solid blue]{ \begin{align*}
\textbf{Power Rule:} & \\[4px] \text{If }f(x) &= x^n, \text{ then } f'(x) = nx^{n-1} \\[4px] \text{for } &\textit{any} \text{ real value of }n.
\end{align*} }\] You can see the graph of $f(x) = x^n$ and its derivative $f'(x) = nx^{n-1}$ in the following Exploration.

Let’s revisit a problem from Chapter 1, where we found the linear approximation to $\sqrt{16.2}.$ At that point, we had to simply tell you the rate at which the function changes at $x = 16,$ but now you can find that rate for yourself!

On the next screen we’ll add to our toolkit of derivatives we can quickly compute, by examining the exponential function $a^x.$

The Upshot