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Warm-up
1. Find the derivative
f ( x)  x  4 x  3
2
2. Find the derivative at the following point.
f ( x)  x  1 at x=3
Table of Contents
• 10. Section 3.2 The Derivative as a
Function
The Derivative as a
Function
• Essential Question – What rules of
differentiation will make it easier to
calculate derivatives?
Notation for Derivative
dy
d
f ( x) or y or
or
f ( x)
dx
dx
'
'
• If derivative exists, we say it is
differentiable
Power Rule
• Power Rule
n
n 1
f '( x )  n x
• Bring down the exponent and
subtract one from the exponent
dy 3
Example ( x )  3x 2
dx
Example -
dy
Example dx
dy
2 1/3
( x 2/3 ) 
x
dx
3
dy 1/ 2
1 1/ 2
1
x 
x

x

dx
2
2 x
More notation
•
d 4
x
dx
x4
means find the derivative of
when x = -2
x 2
2 more rules
• Constant multiple
Example - y=4x
2
(cf )  cf
'
'
y '  4(2 x)  8x
• Sum and Difference
( f  g)  f  g
'
Example - f '(2 x  3x)  4 x  3
2
'
'
Differentiating a
polynomial
dp
5
3
2
Find
if p  t  6t  t  16
dt
3
dp
5
2
= 3t  12t   0
dt
3
Derivative of ex
d x
x
e e
dx
Example
• Find the equation of the tangent line
to the graph of f(x) = 3ex -5x2 at x=2
f ' ( x)  3e x  10 x
f ' (2)  3e2  10(2)  2.17
f (2)  3e2  5(2)2  2.17
y  2.17  2.17( x  2)
y  2.17 x  2.17
What information does the
derivative at a point tell us?
• Tells us whether the tangent line has
a positive or negative slope
• Tells us how steep the line is (the
larger the derivative, the steeper
the line)
• Tells us if there is a turning point
(slope is 0)
Horizontal Tangents
• Does y = x4 – 2x2 + 2 have any
horizontal tangents?
• First find the derivative, then set = 0
(because the slope of a horizontal
line is 0) dy
= 4 x3  4 x  0
dx
4 x( x 2  1)  0
4x  0
( x2  1)  0
x0
x  1
Calculator example
• Find the points where horizontal
tangents occur.
s(t )  0.2t  0.7t  2t  5t  4
ds
 0.8t 3  2.1x 2  4 x  5  0
dt
On calculator, find zeros
4
3
2
t  1.862, 0.9484, 3.539
Plug these back into O.F. to get points
Graphing f’(x) from f(x)
• Find slope at each point
• Make a new graph using same x points
and the slope as the y point
• If f is increasing, f ‘ will be positive
(above the x axis)
• If f has a turning point, f ‘ will be 0
• If f is decreasing, f ‘ will be negative
(below the x axis)
Graph example
• Given the graph of f(x), which of A
or B is the derivative?
Differentiability
• Differentiability implies continuity
• If f is differentiable at x = c, then f
is continuous at x = c
• The opposite is not true
• A function can be continuous at x = c,
but not differentiable
4 times a derivative fails to
exist
• Corner
• Cusp
4 times a derivative fails to
exist
• Vertical tangent
• Discontinuity
Local Linearity
• A function that is differentiable
closely resembles its own tangent line
when viewed very closely
• In other words, when zoomed in on a
few times, a curve will look like a
straight line.
Example
yx
2
y x
Assignment
• Pg. 139 #1-11 odd, 17-29 odd, 47-57
odd, 71-75 odd, 76-80 all, 83-87 odd