2.1 DERIVATIVES

We are now ready to explain what is meant by the slope of a curve or the velocity of a moving point. Consider a real function f and a real number a in the domain of f. When x has value a, f(x) has value f(a). Now suppose the value of x is changed from a to a hyperreal number a+Δx which is infinitely close to but not equal to a. Then the new value of f(x) will be f(a+ Δx). In this process the value of x will be changed by a nonzero infinitesimal amount Δx, while the value of f(x) will be changed by the amount

f(a +Δ x) - f(a).

The ratio of the change in the value of f(x) to the change in the value of x is

f(a +Δ x) - f(a)

Δ x

The ratio is used in the definition of the slope of f which we now give.

DEFINITION

S is said to be the slope of f at a if

for every nonzero infinitesimal x.

The slope, when it exists, is infinitely close to the ratio of the change in f(x) to an infinitely small change in x. Given a curve y=f(x), the slope of f at a is also called the slope of the curve y=f(x) at x=a. Figure 2.1.1 shows a nonzero infinitesimal Δx and a hyperreal straight line through the two points on the curve at a and a+Δx. The quantity

is the slope of this line, and its standard part is the slope of the curve.

Figure 2.1.1

The slope of f at a does not always exist. Here is a list of all the possibilities.

(1) the slope of f at a exists if the ratio

is finite and has the same standard part for all infinitesimal Δx ≠ 0. It has the value

(2) the slope of f at a can fail to exist in any of four ways:

(a) f(a) is undefined.

(b) f(a+Δ x) is undefined for some infinitesimal Δx ≠ 0.

(c) the term _____________ is infinite for some infinitesimal Δx ≠ 0.

(d) the term ____________ has different standard parts for different

infinitesimals Δx ≠ 0.

We can consider the slope of f at any point x, which gives us a new function of x.

DEFINITION

Let f be a real function of one variable. The derivative of f is the new function f ' whose value at x is the slope of f at x. In symbols,

Whenever the slope exists.

The derivative f ' (x) is undefined if the slope of f does not exist at x.

For a given point a, the slope of f at a and the derivative of f at a are the same thing. We usually use the word “slope” to emphasize the geometric picture and “derivative” to emphasize the fact that f ' is a function.

The process of finding the derivative of f is called differentiation. We say that f is differentiable at a if f ' (a) is defined; i.e., the slope of f at a exists.

Independent and dependent variables are useful in the study of derivatives. Let us briefly review what they are. A system of formulas is a finite set of equations and inequalities. If we are given a system of formulas which has the same graph as a simple equation y=f(x), we say that y is a function of x, or that y depends on x, and we call x the independent variable and y the dependent variable.

When y=f(x), we introduce a new independent variable Δx and a new dependent variable Δy, with the equation

(1) Δy= f(x+ Δx) - f(x).

This equation determines Δy as a real function of the two variables x and Δx, when x and Δx vary over the real numbers. We shall usually want to use the Equation 1 for Δy when x is a real number and Δx is nonzero infinitesimal. The Transfer Principle implies that Equation 1 also determines Δy as a hyperreal function of two variables when x and Δx are allowed to vary over the hyperreal numbers.

Δy is called the increment of y. Geometrically, the increment Δy is the change in y along the curve corresponding to the change Δx in x. The symbol y' is sometimes used for the derivative, y' = f ' (x). Thus the hyperreal equation

now takes the short form

The infinitesimal Δx may be either positive or negative, but not zero. The various possibilities are illustrated in Figure 2.1.2 using an infinitesimal microscope. The signs of Δx and Δy are indicated in the captions.

Our rules for standard parts can be used in many cases to find the derivative of a function. There are two parts to the problem of finding the derivative f ' of a function f:

(1) find the domain f '.

(2) Find the value of f '(x) when it is defined.

Figure 2.1.2

EXAMPLE 1 Find the derivative of the function

f(x) = x3.

In this and the following examples we let x vary over the real numbers and Δx vary over the nonzero infinitesimals. Let us introduce the new variable y with the equation y= x3. We first find Δy/Δx.

Next we simplify the expression for Δy/Δx.

Then we take the standard part,

Therefore,

We have shown that the derivative of the function

f(x) =x3

is the function f '(x) =3x2

with the whole real line as domain. f (x)and f '(x) are shown in Figure 2.1.3.

Figure 2.1.3

EXAMPLE 2 Find f '(x) given f(x) =_____

Case 1 x<0. Since ____ is not defined, f '(x) does not exist.

Case 2 x=0. When x is a negative infinitesimal, the term

is not defined because ____is undefined. When Δx is a positive infinitesimal, the term

is defined but its value is infinite. Thus for two reasons, f '(x) does not exist.

Case 3 x >0. Let y=____. Then

We then make the computation

Taking standard parts,

Therefore, when x >0, f '(x) = __________.

So the derivative of f(x) = _________.

is the function f '(x) = __________,

and the set of all x >0 is its domain (see Figure 2.1.4)

Figure 2.1.4

EXAMPLE 3 Find the derivative of f(x) = 1/x.

Case 1 x=0. Then 1/x is undefined so f '(x) is undefined.

Case 2 x ≠ 0.

Simplifying,

Taking the standard part,

Thus f '(x) = -1/x²

The derivative of the function f(x) = 1/ x is the function f '(x) = -1/x²

Whose domain is the set of all x ≠ 0. Both functions are graphed in Figure 2.1.5.

Figure 2.1.5

EXAMPLE 4 Find the derivative of f(x) = |x|.

Case 1 x >0. In this case |x|=x, and we have

y=x,
y+ Δy= x+Δx,

Δy= Δx,

______ = 1, f '(x) =1

Case 2 x<0. Now |x| = -x, and

y= -x,
y+ Δy= -(x+Δx,)

Δy= -(x+Δx) - (-x) = - Δx,

______ = -1, f '(x) = -1.

Case 3 x=0. Then

and

The standard part of y/x is then 1 for some values of x and -1 for others. Therefore f '(x) does not exist when x=0.

in summary,

Figure 2.1.6 shows f(x) and f '(x).

Figure 2.1.6

The derivative has a variety of applications to the physical, life, and social sciences. It may come up in one of the following contexts.

Velocity: If an object moves according to the equation s= f(t) where t is time and s is distance, the derivative v= f '(t) is called the velocity of the object is time t.

Growth rates: A population y (of people, bacteria, molecules, etc.) grows according to the equation y= f(t) where t is time. Then the derivative y' = f '(t) is the rate of growth of the population y at time t.

Marginal values (economics ): Suppose the total cost (or profit, etc.) of producing x items is y=f(x) dollars. Then the cost of making one additional item is approximately the derivative y '= f '(x) because y' is the change in y per unit change in x. This derivative is called the marginal cost.

EXAMPLE 5 A ball thrown upward with initial velocity b ft per sec will be at a height

y = bt - 16t²

feet after t seconds. Find the velocity at time t. Let t be real and Δt ≠ 0,

Infinitesimal.

At time t sec, v= y' = b- 32t ft/sec.

Both functions are graphed in Figure 2.1.7.

Figue 2.1.7

EXAMPLE 6 Suppose a bacterial culture grows in such a way that at time t there are t 3

bacteria. Find the rate of growth at time t= 100 sec.

y= t 3 y'= t 3 by Example 1.

At t = 1000, y'=3,000,000 bacteria/sec.

EXAMPLE 7 Suppose the cost of making x needles is _____ dollars. What is the marginal cost after 10,000 needles have been made?

y=______, y' =_______ by Example 2.

At x= 10,000, y' =_______ =______ dollars per needle.

Thus the marginal cost is one half of a cent per needle.

PROBLEMS FOR SECTION 2.1

Find the derivative of the given function in Problems 1-21.

1 f(x) =x² 2 f(t) = t² + 3

3 f(x) =1-2x² 4 f(x) =3x² + 2

5 f(t) =4t 6 f(x) =2-5x

7 f(t) =4t3 8 f(t) = -t3

9 f(u) =_____ 10 f(u) =_____

11 g(x)=______ 12 g(x)=______

13 g(t)=______ 14 g(x)= t -3

15 f(y)=3y-1 + 4y 16 f(y)=2y3 + 4y²

17 f(x)=ax + b 18 f(x)=ax²

19 f(x)=________ 20 f(x)=1/(x+2)

21 f(x)=1/(3-2x)

22  Find the derivative of f(x) = 2x² at the point x=3.

23  Find the slope of the curve f(x) = ________ at the point x=5.

24  An object moves according to the equation y= 1/(t+2), t ≥ 0.

    Find the velocity as a function of t.

25  A particle moves according to the equation y=t4. Find the velocity as a function of t.

26  Suppose the population of a town grows according to the equation y= 100t + t2. Find the

    rate of growth at time t= 100 years.

27  Suppose a company makes a total profit of 1000x- x² dollars on x items. Find the

    Marginal profit in dollars per item when x = 200, x=500, and x=1000.

28  Find the derivative of the function f(x) = |x + 1|.

29  Find the derivative of the function f(x) = |x 3|.

30  Find the slope of the parabola y= ax² + bx + c where a, b, c are constants.