Math 135 Business Calculus Spring 2009Class Notes1.4 Differentiation Using Limits of Difference Quotients

In the previous section, we saw that a secant line joining the points°x, f(x)

¢and

°x + h, f(x + h)

¢on

the graph of a function y = f(x) has slope

msecant =f(x + h)− f(x)

h.

This difference quotient also represents the average rate of change of f(x) over the interval [x, x + h].In this section, we’ll see that by taking the limit of this difference quotient as h → 0, we obtain

the slope of the tangent line to the graph or the instantaneous rate of change of f(x) at x.

TANGENT LINES

A tangent to a curve is sometimes described as a line that touches the curve in exactly one point. Fora circle, results from geometry tell us that a tangent line to a point on the circle is perpendicular to aradius intersecting the circle at that point, as shown in the figure below on the left.

P

L L

M

For more complicated curves, the above description is inadequate. The above figure on the rightdisplays two lines L and M passing through a point P on a curve. The line M intersects only once,but it certainly does not look like what is thought of as a tangent. In contrast, the line L looks like atangent, but it intersects twice.

In order to define what we mean by the tangent line and in order to determine the tangent line,we need to use a limiting process. Suppose we have a function y = f(x) and we want to determine thetangent line at a point P on the curve. We start by drawing a secant line to the curve passing throughP and another point Q on the curve. The secant line provides an approximation to the tangent line.The closer the point Q is to the point P , the better the secant line will approximate the tangent line.

f (x)

Secant line

Tangent line

x

P(x, f (x))

y

Q (x + h, f (x + h))

f (x)

f (x + h)

f (x)

Secant lines

Tangent line

x

P

y

Q1

Q2

Q3

Q4

By letting the point Q slide down the curve towards P , the corresponding secant lines will rotate aboutthe point P . The tangent line will be the limiting position of the secant lines as Q approaches P .

15

16 Chapter 1 Differentiation

The slope of the secant line PQ is given by

msecant =f(x + h)− f(x)

h.

As the point Q slides down the curve towards P , then the x-values x+h approach x, so h approaches 0.

DEFINITION OF SLOPE OF TANGENT LINE

The slope of the tangent line the graph of y = f(x) at the point°x, f(x)

¢is

mtangent = limh→0

f(x + h)− f(x)h

.

This limit is also the instantaneous rate of change of f(x) at x.

The limit in the above definition occurs so widely, it is given a special name and notation.

DEFINITION OF DERIVATIVE

For a function y = f(x), its derivative at x is the function f 0 defined by

f 0(x) = limh→0

f(x + h)− f(x)h

provided that the limit exists. If f 0(x) exists, then we say that f is differentiable at x. The notationf 0(x) is read “f prime of x.”

PROCEDURE FOR CALCULATING A DERIVATIVE

To calculate the derivative of a function y = f(x):

1. Write out the difference quotient:f(x + h)− f(x)

h2. Simplify the difference quotient algebraically. After simplifying, the h in the denominator should

cancel out.3. Compute the limit algebraically using the Limit Principles.

EXAMPLE Let f(x) = x2. Find f 0(x). Then find f 0(−1) and f 0(2). What do these represent onthe graph?

–3 –2 –1 1 2 3 x

–2

–1

1

2

3

4

5y

1.4 Differentiation Using Limits of Difference Quotients 17

EXAMPLE Let f(x) = x3. Find f 0(x). Then find f 0(−1) and f 0(1.5). Find the equation of thetangent line to the graph at x = −1.

–3 –2 –1 1 2 3x

–6

–5

–4

–3

–2

–1

1

2

3

4

5

6y

EXAMPLE Let f(x) = 3/x. Find f 0(x). Then find f 0(2) and the equation of the tangent line to thegraph at x = 2.

–5 –4 –3 –2 –1 1 2 3 4 5x

–5

–4

–3

–2

–1

1

2

3

4

5y

18 Chapter 1 Differentiation

DIFFERENTIABILITY

In the preceding example, note that since f(0) does not exist for f(x) = 3/x, the difference quotient

f(0 + h)− f(0)h

cannot be computed. Therefore f 0(0) does not exist. We say that f is not differentiable at 0.In general, if a function is not defined at a point, then it is not differentiable there. In fact, if a

function is discontinuous at a point, then it is not differentiable at the point. There are other situations,however, in which a function is not differentiable.

EXAMPLE Where is the function f(x) = |x| differentiable?

In general, if a function has a sharp point or corner, then it will not be differentiable at that point.For instance, the following functions are not differentiable at the indicated points.

A function will also fail to be differentiable at a point if it has a vertical tangent. Summarizing, wethen have three situations where a function is not differentiable:

• The graph has a sharp point or corner at that point.• The graph has a vertical tangent line.• The function is discontinuous at that point.