The 9.3 Derivative
The student will learn about: Rate of change Slope of the tangent line The derivative Existence/Nonexistence of the derivative
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Difference Quotient: Slope
The difference quotient that follows gives the average rate of change of the function passing through P and Q:
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slope
3 2
f (a + h)
f (a + h ) ! f (a ) h
f (a + h) f (a)
-1
1
f (a) O (a)
1
(a + h)
2
h
2
Example 1
The revenue is given by R (x) = x (75 3x) for 0 ! x ! 20.
What is the change in revenue if production changes from 9 to 12?
What is the average rate of change in revenue if production changes from 9 to 12?
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The Instantaneous Rate of Change.
Consider the function y = f (x) only at the point (a, f (a)) . The limit of the difference quotient that follows gives the instantaneous rate of change of the function passing through (a, f (a)): The instantaneous rate of change =
f (a + h) " f (a) lim h!0 h
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4
Visual Interpretation
3
f (a + h)
2
Tangent
Slope of tangent. Instantaneous Rate of Change.
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f (a)
-1
lim f (a + h ) ! f (a ) h"0 h
O
(a)
1
(a + h)
2
Let h approach 0
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3
h
Definition of the Derivative
Given y = f (x), the slope of the graph at the point (a, f (a)) is given by
h!0
lim
f (a + h) " f (a) h
Provided that the limit exist. We define it to be the derivative of f at x, denote it by f (x). We define the tangent line to y=f(x) at the point (a, f (a)) to be the line through this point of slope equal to f (x). f (x) is the instantaneous rate of change at x (e.g. velocity). If f (x) exists for each x in the open interval b), (a, then f is said to be differentiable over (a, b).
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Four-Step Process
To find f (x) we use a four-step process Step 1. Find f (x = h) Step 2. Find f (x + h) f (x) Step 3. Find
f ( x + h) ! f ( x) h
Step 4. Find lim
f ( x + h) ! f ( x) h h"0
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Example 2
Find the derivative of f (x) = x 2 3x f (x + h) = (x + h) 2 3(x + h) f (x + h) f (x)
f ( x + h) ! f ( x) h
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Example 3
Find the slope of the graph of f (x) = x 2 3x at x = 0, x = 2, and x = 3. From example 2 we found the derivative of this function at x to be f (x) = 2x - 3
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Example 4
R(x) = 60x .02x2 Find R (x)
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Nonexistence of the Derivative.
The existence of a derivative at x = a depends on the existence of a limit at x = a; that is, on the existence of
f (a) =
f (a + h ) ! f (a ) h h"0 lim
If the limit does not exist at x = a, we say that the function is nondifferentiable at x = a, or f (x) does not exist.
f (x) = x |x+h|"|...