The Secant-Line Calculation of the Derivative

We have said that the derivative gives the slope of the tangent line and presented formulas for calculating the derivative for various curves.

However, we have not discussed how these formulas were derived.

The fundamental idea for calculating the slope of the tangent line at a point P is to approximate the tangent line very closely by secant lines.

A secant line at P is a straight line passing through P and a nearby point Q on the curve.

As we take Q closer to P, the accuracy with which the slope of the secant line approximates the slope of the tangent line increases.

Suppose that the point P is (x, f(x)) and that Q is h horizontal units away from P. Q will have x-coordinate x+h, and y-coordinate f(x+h). Recall that the slope of a line is given by

the slope of a line through points (x1,y1) and

(x2, y2) =

12

12

xx

yy

Then slope of the secant line through the points P and Q is

slope of secant line =

h

xfhxf

xhx

xfhxf )()(

)(

)()(

In order to improve the accuracy of our secant line approximation of the tangent line, we move Q closer to P by letting h approach 0.

Then, the slope of the secant line approaches the slope of the tangent line and

h

xfhxf )()( approaches f’(x)

To calculate f’(x) using a secant line approximation, calculate the difference quotient

h

xfhxf )()(

Then, let h approach zero. The quantity

h

xfhxf )()( will approach f’(x).

Verification of the Power Rule for r = 2Let f(x) = x2. By the Power Rule, f’(x) = 2x. By using the secant line approximation, we have

slope of secant line =

h

xfhxf )()(

h

xhx 22)(

h

xhxhx 222 )2(

hxh

hxh

h

hxh

2

)2(2 2

As h approaches 0, 2x + h approaches 2x.

1.4 Limits and the Derivative

The notion of a limit is a fundamental idea of calculus.

Although we did not explicitly state it, we were using the concept of a limit in the previous section.

Using the geometric reasoning from the previous section, we have the following procedure for calculating the derivative of a function f(x) at x = a.

Calculate the difference quotient

h

afhaf )()(

Let h approach zero by allowing it to assume both positive and negative values arbitrarily close to but different from zero.

The quantity h

afhaf )()(

will approach f’(a).

We say that the number f’(a) is the limit of the difference quotient as h approaches zero and we write this as

h

afhafaf

h

)()(lim)('

0

Consider the case where f(x) = x2 at x = 2.

The difference quotient has the form

h

h

h

fhf 22 2)2()2()2(

Letting h approach zero from the positive side shows that the difference quotient approaches 4.

As does letting h approach zero from the negative side

In other words, 4 is the limit of the difference quotient as h approaches zero

42)2(

lim22

0

h

hh

Since the value of the difference quotient approaches the derivative f’(2), we conclude that f’(2) = 4.

We say that f is differentiable at x = a if

h

afhaf )()(

approaches some number as h approaches zero.

If the difference quotient does not approach any specific number as h approaches zero, then we say f is nondifferentiable at x = a.

We can look at limits in a more general way.

Let g(x) be a function and a be a number. We say that L is the limit of g(x) as x approaches a provided that g(x) can be made arbitrarily close to L by taking x sufficiently close (but not equal) to a. We write this as

Lxgax

)(lim

If, as x approaches a, g(x) does not approach a specific number, we say that the limit of g(x) as x approaches a does not exist.

Determine )53(lim2

xx

We can examine the values of 3x – 5 as x approaches 2 from both sides

As x approaches 2, we see that 3x – 5 approaches 1.

We conclude that 1)53(lim2

xx

Determine if

)(lim2xg

x

exists for the functions in the following graphs

Limit Theorems

Suppose that

)(lim xfax

and )(lim xg

ax

both exist.

Then we have the following theorems.

Examples

3

2lim xx

)155(lim 3

2

x

x

5

3

2

155lim

x

xx

Limit of a Polynomial Function

Let p(x) be a polynomial function and a any number. Then

)()(lim apxpax

Limit of a Rational Function

Let r(x) = p(x)/q(x) be a rational function where p(x) and q(x) are polynomials. Let a be a number such that q(a) does not equal zero. Then,

)()(lim arxrax

The basic rules of differentiation are obtained from the limit definition of the derivative.

There are three main steps for calculating the derivative of a function f(x) at x = a.

1. Write the difference quotient h

afhaf )()(

2. Simplify the difference quotient.

3. Find the limit as h approaches zero.

Infinity and Limits

Consider the function f(x) whose graph is

As x increases, the value of f(x) approaches 2.

In this case, we say that the limit of f(x) as x approaches infinity is 2.

We express this using limit notation as

2)(lim

xfx

Similarly, if we examine the following graph, we will note that as x grows large in the negative direction, the value of f(x) approaches 0.

We express this using limit notation as

0)(lim

xfx

Example

1

1lim

2xx