Definition of the Derivative
Lesson 3.4
Tangent Line
Recall from geometry
• Tangent is a line that touches the circle at only
one point
Let us generalize the concept to functions
• A tangent will just "touch" the line but not pass
through it
• Which of the above lines are tangent?
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A Secant Line
Crosses the curve twice
x=a
h
The slope of the secant will be
y f (a  h)  f (a ) f (a  h)  f (a )


x
aha
h
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Tangent Line
Now let h get smaller and smaller
x=a
h
Try this with the TI
Nspire again
f (a  h)  f (a )
lim
 The slope of the tangent line
h 0
h
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Tangent Line
Try it out …given
f ( x)  6 x 2  4 x
• Determine the slope of the tangent line at x = 0
Evaluate
f (0  h)  f (0)
lim
h 0
h
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Tangent Line
Use the limit command on your calculator
to determine the slope of f ( x)  6 x 2  4 x
for x = 1
Once you have the slope and the point
• You can determine the equation of the line
y  y1  m  ( x  x1 )
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The Derivative
We will define the derivative of f(x) as
f ( a  h)  f (a )
f '( x)  lim
h 0
h
Note
• The derivative is the rate of change function
for f(x)
• The derivative is also a function of x
• The limit must exist
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Comparison
Difference Quotient
f (a )  f (b)
a b
Derivative
lim
h 0
f (a  h)  f (a )
h
Slope of secant
Slope of tangent
Average rate of
change
Average velocity
Instantaneous rate of
change
Instantaneous velocity8
Finding f'(x) from Definition
1.
2.
Strategy
Find f(x + h)
Find and simplify f(x + h) – f(x)
3.
Divide by h to get
4.
Let h → 0
f ( x  h)  f ( x )
h
f ( x  h)  f ( x )
f '( x)  lim
h 0
h
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Try It Out
Use the strategy to find the derivatives of
these functions
5
f ( x) 
x
f ( x)  4 x  2 x
3
w( x)  x  12
Hint: rationalize the
numerator
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Equation of the Tangent Line
We stated previously that once we
determine the slope of the tangent
y  y1  m  ( x  x1 )
We can "cut to the chase" and state it as
y  y1  f '( x)  ( x  x1 )
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Warning
Our definition of derivative included
the phrase "if the limit exists"
Derivatives do not exist at "corners" or
"sharp points" on the graph
• The slope is different on each side of the point
f(x) = | x – 3 |
• The limit does not exist
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Assignment
Lesson 3.4
Page 210
Exercises 1 – 51 odd
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