2-2: differentiation rules objectives: learn basic differentiation rules explore relationship...
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2-2: Differentiation RulesObjectives:•Learn basic differentiation rules•Explore relationship between derivatives and rates of change
© 2002 Roy L. Gover (www.mrgover.com)
Basic Differentiation Rules
1.Constant Rule
The derivative of a constant function is 0. There is no rate of change.
[ ]d
c odx
Let’s prove it.
Duh…
Find the following derivatives using the definition
𝑓 (𝑥 )=𝑥𝑓 (𝑥 )=𝑥2
𝑓 (𝑥 )=𝑥3Make a guess about
Find the following derivatives using the definitionMake a guess about
Basic Differentiation Rules
1[ ]n ndax nax
dx
2. Power Rule
Where a & n are real numbers and n is rational
Warm-UpFind the derivative using the definition (sometimes called the limit of the difference quotient) and the power rule. Confirm that you get the same answer.
2( ) 2f x x
Basic Differentiation Rules
[ ( )] ( )d d
cf x c f xdx dx
3. Constant Multiple Rule
Where c is a constant
Basic Differentiation Rules
[ ( )] '( )d
cf x cf xdx
3. Constant Multiple Rule
Note: alternate notation
( )d
f xdx
'( )f xis the same as
Basic Differentiation Rules
3. Sum or Difference Rule
( ) ( ) '( ) '( )d
f x g x f x g xdx
The derivative of the sum or difference is the sum or difference of the derivatives
Basic Differentiation Rules
4. Derivatives of Sine & Cosine
sin cosd
x xdx
cos sind
x xdx
Prove it! Now!
4. Derivatives of Sine & Cosine
sin cosd
x xdx
cos sind
x xdx
ExampleFind the derivative, if it exists:
1. 3
2. ( ) 2
3. ( ) 2
y
f x
h t
We used what rule(s)?
ExampleFind the derivative, if it exists:
2
5
1.
2. ( )
3. ( ) 3
y x
f x x
h t t
We used what rule(s)?
ExampleFind the derivative, if it exists:
3 2
11.
2. ( )
yx
f x x
We used what rule(s)?
ExampleFind the derivative, if it exists:
3
31.
25
2. ( )2
xy
f xx
We used what rule(s)?
Try ThisFind the derivative:
2
1 y
x
33
22
dyx
dx x
First write as then use the power rule:
2x
Try ThisFind the derivative:
3
2
dy x
dx
3 ( )f x xFirst write as_____, then…
Try ThisFind the derivative:
27(9 ) 126dy
x xdx
2
7 ( )
(3 )f x
x First write as_____, then…
Example Find the slope of
3( )f x xat x=
x=2
2-1
x=-1
Review
1 1( )y y m x x
Point-Slope Form:
1 at dy
x xdx
1( )f x
Review
y mx b
Slope-y intercept Form:
Example
x=2
Find the equation of line tangent to at x=2
2( )f x x
Try This Find the equation of line tangent to at x=1
3( )f x x
x=13 2y x
ExampleFind the derivative, if it exists:
3( ) 4 5f x x x
We used what rule(s)?
Try ThisFind the derivative, if it exists:
43( ) 3 2 2
2
xh x x x
3 2'( ) 2 9 2h x x x
ExampleFind the derivative, if it exists:
sin1.
2
2. ( ) 2cos
xy
f
Important Idea
The slope of the sin function at a point is the value of the cos function at the point
3
ExampleIf an object is dropped, its height above the ground is given by 2( ) 16 200s t t
1. Find the average velocity between 1 and 3 seconds.2. Find the instantaneous velocity at 3 seconds.
ExampleIf an object is dropped, its height above the ground is given by 2( ) 16 200s t t
1. Find the average velocity between 1 and 3 seconds.
ExampleIf an object is dropped, its height above the ground is given by 2( ) 16 200s t t 2. Find the instantaneous velocity at 3 seconds.
Lesson Close
This lesson demonstrated the use of several differentiation rules. There will be others in future lessons. You must memorize these rules.
Assignment
123/1-47 odd & 71