Basic Differentiation Rules and Rates of Change

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Basic Differentiation Rules and Rates of Change

• Section 2.2

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After this lesson, you should be able to

• Find the derivative using the Constant Rule.
• Find the derivative using the Power Rule.
• Find the derivative using the Constant Multiple
  Rule and the Sum and Difference Rules.
• Find the derivative of sine and cosine.
• Use derivatives to find rates of change.

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Slope of Tangent Line at (x, f(x))
Q
P
Let
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Rules For Computing Derivatives
Constant Rule
Power Rule
Ex
Ex
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Rules For Computing Derivatives
Ex
Ex
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Rules For Computing Derivatives
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Examples
Example
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Examples
Example Find the derivative of
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Examples
Example Find the derivative of
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Finding the Equation of a Tangent Line
Example Find the equation of the tangent line
at x1 and take a peek at the graph.
Pt f(1)
(1,2)
peek at the graph
slope f(x)
5x4 6x
f(1)
-1
equation
y 2 (-1)(x 1)
y -x 3
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Finding an Equation of a Horizontal Tangent Line
Example Find an equation for the horizontal
tangent line to
peek at the graph
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Rates of Change
Some Examples population growth rates,
production rates, velocity and acceleration
Common example is the motion of an object along a
straight line.
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Motion Along a Straight Line
s position
S(t) position function
S (ft)
S(t?t)
Ave. velocity during the time interval from t to
(t?t)
S(t)
t (sec)
t
(t?t)
Instantaneous velocity at time t
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Motion Along a Straight Line
v(t) gt 0,
v(t) lt 0,
v(t) 0,
stopped instantaneously
rate of change of position
s(t) position (ft)
rate of change of velocity
v(t) s(t) velocity (ft/sec)
a(t) v(t) s(t) accel (ft/s2)
speed v(t)
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Free Falling Object
gravity
initial velocity
initial position
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Example of Free Falling Object
Example A ball is thrown straight down from the
top of a 220-foot building with an initial
velocity of 22 feet/second. What is its
velocity after 3 seconds? What is its velocity
after falling 108 feet?
220
Describes the motion of the object
0
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Example of Free Falling Object
Example A ball is thrown straight down from the
top of a 220-foot building with an initial
velocity of 22 feet/second. What is its
velocity after 3 seconds? What is its velocity
after falling 108 feet?
220
Describes the motion of the object
0
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Homework
Section 2.2 page 113 1-51 odd, 53ab, 55ab, 57,
61, 81-86 all
Rate of change problems 92 and 93