Section 3.1 Notes: Derivative of a Function

Derivatives

Acetaminophen is a pain reliever but toxic in large doses.
A study found that only 30% of parents could accurately calculate the correct dosage (mg) of acetaminophen for their children.
Dosage calculation rule for children ages 1 to 12: $D(t) = \frac{750t}{t + 12}$ , where t is age in years.

3.1 Derivative of a Function

Overview

Chapter 2 covered finding the slope of a tangent to a curve as the limit of slopes of secant lines.
Example 4 in Section 2.4 derived a formula for the slope of the tangent at an arbitrary point a, $(\frac{-1}{a^2})$ on the graph of the function $f(x) = \frac{1}{x}$ .
Differential calculus is the study of rates of change of functions.
The derivative is a 17th-century breakthrough that enabled mathematicians to unlock the secrets of planetary motion and gravitational attraction.

Definition of Derivative

The derivative is key to modeling instantaneous change mathematically.
The slope of a curve $y = f(x)$ at the point where $x = a$ is:
$m = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$

DEFINITION Derivative

The derivative of the function f with respect to the variable x is the function $f'(x)$ whose value at x is:
$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$ ,
provided the limit exists.
The domain of $f'$ , the set of points in the domain of f for which the limit exists, may be smaller than the domain of f.
If $f'(x)$ exists, f has a derivative (is differentiable) at x.
A function that is differentiable at every point of its domain is a differentiable function.

EXAMPLE 1 Applying the Definition

Differentiate $f(x) = x^3$ . Applying the definition:
$f'(x) = \lim{h \to 0} \frac{f(x + h) - f(x)}{h} = \lim{h \to 0} \frac{(x + h)^3 - x^3}{h}$
$= \lim{h \to 0} \frac{x^3 + 3x^2h + 3xh^2 + h^3 - x^3}{h} = \lim{h \to 0} \frac{3x^2h + 3xh^2 + h^3}{h}$
$= \lim_{h \to 0} (3x^2 + 3xh + h^2) = 3x^2$

DEFINITION (ALTERNATE) Derivative at a Point

The derivative of the function f at the point $x = a$ is the limit
$f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$ ,
provided the limit exists.
After finding the derivative of f at a point using the alternate form, we can find the derivative of f as a function by applying the resulting formula to an arbitrary x in the domain of f.

EXAMPLE 2 Applying the Alternate Definition

Differentiate $f(x) = \sqrt{x}$ using the alternate definition.
At the point $x = a$ ,

$f'(a) = \lim{x \to a} \frac{f(x) - f(a)}{x - a} = \lim{x \to a} \frac{\sqrt{x} - \sqrt{a}}{x - a} = \lim_{x \to a} \frac{\sqrt{x} - \sqrt{a}}{x - a} \cdot \frac{\sqrt{x} + \sqrt{a}}{\sqrt{x} + \sqrt{a}}$

$= \lim{x \to a} \frac{x - a}{(x - a)(\sqrt{x} + \sqrt{a})} = \lim{x \to a} \frac{1}{\sqrt{x} + \sqrt{a}} = \frac{1}{2\sqrt{a}}$

Applying this formula to an arbitrary x > 0 in the domain of f identifies the derivative as the function $f'(x) = \frac{1}{2\sqrt{x}}$ with domain $(0, \infty)$ .

Notation

There are many ways to denote the derivative of a function $y = f(x)$ .
- $f'(x)$ .
- $y'$ “y prime” Nice and brief, but does not name the independent variable.
- $\frac{dy}{dx}$ “dy dx” or “the derivative of y with respect to x” Names both variables and uses d for derivative.
- $\frac{df}{dx}$ “df dx” or “the derivative of f with respect to x” Emphasizes the function’s name.
- $\frac{d}{dx}f(x)$

Relationships Between the Graphs of f and f'

When we have the explicit formula for $f(x)$ , we can derive a formula for $f'(x)$ using methods like those in Examples 1 and 2.
We can think of the derivative at a point in graphical terms as slope, we can get a good idea of what the graph of the function $f'(x)$ looks like by estimating the slopes at various points along the graph of f.

EXAMPLE 3 Graphing f' from f

Graph the derivative of the function f whose graph is shown in Figure 3.3a. Discuss the behavior of f in terms of the signs and values of $f'(x)$ .
The slope at point A of the graph of f in part (a) is the y-coordinate of point A' on the graph of $f'(x)$ in part (b), and so on.
In particular, notice that f is decreasing where $f'(x)$ is negative and increasing where $f'(x)$ is positive.
Where $f'(x)$ is zero, the graph of f has a horizontal tangent, changing from increasing to decreasing at point C and from decreasing to increasing at point F.

EXPLORATION 1 Reading the Graphs

Suppose that the function f in Figure 3.3a represents the depth y (in inches) of water in a ditch alongside a dirt road as a function of time x (in days).
1. What does the graph in Figure 3.3b represent? What units would you use along the y-axis?
2. Describe as carefully as you can what happened to the water in the ditch over the course of the 7-day period.
3. Can you describe the weather during the 7 days? When was it the wettest? When was it the driest?
4. How does the graph of the derivative help in finding when the weather was wettest or driest?
5. Interpret the significance of point C in terms of the water in the ditch. How does the significance of point C' reflect that in terms of rate of change?
6. It is tempting to say that it rains right up until the beginning of the second day, but that overlooks a fact about rainwater that is important in flood control. Explain.

EXAMPLE 4 Graphing f from f'

Sketch the graph of a function f that has the following properties:
- $f(0) = 0$
- the graph of $f'(x)$ , the derivative of f, is as shown in Figure 3.4
- f is continuous for all x.
To the left of $x = 1$ , the graph of f has a constant slope of $-1$ ; therefore we draw a line with slope $-1$ to the left of $x = 1$ , making sure that it goes through the origin.
To the right of $x = 1$ , the graph of f has a constant slope of 2, so it must be a line with slope 2.

Graphing the Derivative from Data

Discrete points plotted from sets of data do not yield a continuous curve, the shape and pattern of the graphed points can be meaningful nonetheless.
It is often possible to fit a curve to the points using regression techniques. If the fit is good, we could use the curve to get a graph of the derivative visually, as in Example 3.
However, it is also possible to get a scatter plot of the derivative numerically, directly from the data, by computing the slopes between successive points, as in Example 5.

EXAMPLE 5 Estimating the Probability of Shared Birthdays

Suppose 30 people are in a room. What is the probability that two of them share the same birthday? Ignore the year of birth.
- The probability of a shared birthday among 30 people is at least 0.706.
- The probabilities grow slowly at first, then faster, then much more slowly past $x = 45$ .

One-Sided Derivatives

A function $y = f(x)$ is differentiable on a closed interval [a, b] if it has a derivative at every interior point of the interval, and if the limits [the right-hand derivative at a] and [the left-hand derivative at b] exist at the endpoints.
- $\lim_{h \to 0^+} \frac{f(a + h) - f(a)}{h}$ [the right-hand derivative at a]
- $\lim_{h \to 0^-} \frac{f(b + h) - f(b)}{h}$ [the left-hand derivative at b]
In the right-hand derivative, h is positive and $b + h$ approaches a from the right. In the left-hand derivative, h is negative and $a + h$ approaches b from the left (Figure 3.8).
Right-hand and left-hand derivatives may be defined at any point of a function’s domain.
A function has a (two-sided) derivative at a point if and only if the function’s right-hand and left-hand derivatives are defined and equal at that point.

EXAMPLE 6 One-Sided Derivatives Can Differ at a Point

Show that the following function has left-hand and right-hand derivatives at $x = 0$ , but no derivative there (Figure 3.9).
$y = \begin{cases} x^2, & x \leq 0 \ 2x, & x > 0 \end{cases}$
The existence of the left-hand derivative:
$\lim{h \to 0^-} \frac{(0 + h)^2 - 0^2}{h} = \lim{h \to 0^-} \frac{h^2}{h} = 0$
We verify the existence of the right-hand derivative:
$\lim{h \to 0^+} \frac{2(0 + h) - 0}{h} = \lim{h \to 0^+} \frac{2h}{h} = 2$
Since the left-hand derivative equals zero and the right-hand derivative equals 2, the derivatives are not equal at $x = 0$ . The function does not have a derivative at 0.