Applied Mathematics for Business and Economics

Lecture Note Differentiation

Chapter 2

Differentiation: Basic Concepts

1 The Derivative

Definition

For the functionyfx= (), the derivative of f at x is defined to be:

(^) ()
0

()(

lim

x

f xxfx)

fx

Δ→ x

⎡ +Δ − ⎤

′ = ⎢ ⎥

⎣ Δ ⎦

provided that the limit exists.

To find a derivative by using the definition

1. Form the ratio
   f()(xxfx)
   x

+Δ −

Δ

, called the difference quotient.

2. Simplify the difference quotient algebraically.


3. Calculate ()
   0

()()

lim

x

f xxfx

fx

Δ→ x

′ ⎡ +Δ − ⎤

= ⎢ ⎥

⎣ Δ ⎦

Example 1

Use the definition of derivative to find f′(x) forf(xx)=^2.

Solution

Step 1: Form the deference quotient.

()()()

(^22)
f xxfx xx x
x x

+Δ − +Δ −

=

ΔΔ

Step 2: Simplify the difference quotient

() () ()

(^22222)
22
2
x x x x xx x x xx x
2
x x
x xx

+Δ − + Δ + Δ − Δ + Δ

==

ΔΔΔ

=+Δ

Step 3: Find the limit.

Δ→lim 2x 0 ( x+Δ = + =xx)^2 0 2x^

Therefore, f′(xx)= 2.

Example 2
Suppose a manufacturer’s profit from the sale of radios is given by the function

Px()= 4 x is the price at which the radios are sold. Find the

selling price that maximizes profit.

00 15( )( )−−x x 2 , where

Solution

Your goal is to find the value of x that maximizes the profitPx( ). This is the value of

x for which the slope of the tangent line is zero. Since the slope of the tangent line is

given by the derivative, begin by computingPx′( ).For simplicity; apply the definition

of the derivative to the unfactored form of the profit function.
Px()=− 400 x^2 +6, 800x−12, 000

You find that