Section 2-3-1

Chapter 2
Models of Growth: Rates of Change

2.3 Symbolic Calculation of Derivatives:
Polynomial Functions

In the last section we saw that the derivative was a measure of the instantaneous rate of change of a variable. We can calculate the derivative by zooming in on the graph of the function in question. We also saw how to use algebra to find the derivative of functions of the form $k t^{2}$ . In this section we will consider more general polynomial functions.

2.3.1 Derivatives of Power Functions: Initial Cases

Before we tackle the problem of finding the derivative of a polynomial function, we start with the simpler question of finding the derivative of a power function. We know some special cases:

$\frac{d}{d t} k$	$= 0$	If there is no change, the rate of change is zero.
$\frac{d}{d t} k t$	$= k$	The rate of change of a linear function is the slope of its graph.
$\frac{d}{d t} k t^{2}$	$= 2 k t$	Look back at Example 1 in Section 2.2.

What about the general power function $t^{n}$ , where $n$ may be any positive integer? We repeat the cases we know (with $k$ set equal to $1$). Note that $t^{0} = 1$ , and $t^{1} = t$ .

$\frac{d}{d t} 1$	$= 0$
$\frac{d}{d t} t$	$= 1$
$\frac{d}{d t} t^{2}$	$= 2 t$

Let's do one more case by direct calculation of $\Delta s/\Delta t$, where $s = t^{3}$ . We will do this one by writing

Δ s = s (t + Δ t) - s (t)

Here are the steps with justifications:

Algebraic Step		Reason
$\frac{Δ s}{Δ t}$	$= \frac{{(t + Δ t)}^{3} - t^{3}}{Δ t}$	We substituted for $s$.
	$= \frac{[t^{3} + 3 t^{2} (Δ t) + 3 t {(Δ t)}^{2} + {(Δ t)}^{3}] - t^{3}}{Δ t}$	We expanded ${(t + Δ t)}^{3}$ .
	$= \frac{3 t^{2} (Δ t) + 3 t {(Δ t)}^{2} + {(Δ t)}^{3}}{Δ t}$	We removed the brackets: $t^{3} - t^{3} = 0$ .
	$= 3 t^{2} + 3 t (Δ t) + {(Δ t)}^{2}$	We divided through by $Δ t$ .

Now, as $Δ t \to 0$ , the second and third terms on the right also approach zero, so

\frac{d}{d t} t^{3} = 3 t^{2}

Activity 1

Carry out a similar calculation to find the derivative of $t^{4}$ . You may use either the factoring technique (Example 1 in Section 2.2) or the binomial expansion technique just illustrated.
Make a conjecture about the derivative of $t^{n}$ for a general positive $n$.

Comment on Activity 1

Contents for Chapter 2

Chapter 2 Models of Growth: Rates of Change

2.3 Symbolic Calculation of Derivatives: Polynomial Functions

Chapter 2
Models of Growth: Rates of Change

2.3 Symbolic Calculation of Derivatives:
Polynomial Functions