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
- Form the ratio
f()(xxfx)
x
+Δ −
Δ
, called the difference quotient.
- Simplify the difference quotient algebraically.
- 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