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Reminder WebAssign Homework Read the next section on the schedule
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MAT 1221Survey of Calculus
Section 2.2 Some Rules for Differentiation
http://myhome.spu.edu/lauw
Expectations Use pencils Use “=“ signs and “lim” notation correctly Do not “cross out” expressions doing cancelations Do not skip steps – points are assigned to all
essential steps Need the transitional statement: “The equation of
the tangent line at…”
Reminder WebAssign Homework Read the next section on the schedule
Recall
hxfhxfxf
h
)()(lim)(0
The slope of the tangent line of y=f(x) at any given value of x.)
Computation of limits are not efficient. We want to have formulas for the
derivatives without compute limits.
Preview Familiar with the many common
notations for derivatives Familiar with the different basic formulas
for differentiation
Notations The following are common notations for
the derivative y=f(x).
( ), , , ( )dy df x y f xdx dx
Constant Function RuleIf , then
Why?
Cxfy )( ( ) 0 0df x cdx
00lim0lim
lim)()(lim)(
00
00
hh
hh
h
hCC
hxfhxfxf
Constant Function RuleIf , then
Why? 1. Geometric Reason 2. Limit computation
( ) 0 0df x cdx
Cxfy )(
Constant Function RuleCxfy )( ( ) 0 0df x c
dx
x
y
C
x
Example 1
(a) ( ) 3, ( )
(b) 1.7,
(c) e,
f x f x
y y
dyydx
Cxfy )( 0)( xf
Power RuleIf , then(n can be any real number)
nxxfy )( 1)( nnxxf
1n nd x nxdx
Power RuleIf , then
Why? (Can be proved by computing limits) Evidences?
nxxfy )( 1)( nnxxf
2.1 Example 2Find the slope of the tangent line ofAt x=2
2( )f x x
2( )f x x
Tangent linat at 2(2)
xslope f
( ) 2f x x
1
( )
( )
n
n
y f x x
f x nx
Example 2
2
3
(a) , (Memorize this special case!)
(b) ( ) , ( )
(c) ,
y x y
f x x f x
dyy xdx
1
( )
( )
n
n
y f x x
f x nx
Example 21
52
(d) ,
(e) ,
1(f) ,
y x y
dyy xdx
y yx
1
( )
( )
n
n
y f x x
f x nx
Constant Multiple Rule If , then
where is a constant
( )y xk u
( )y k u x
k
( )d dku x k u xdx dx
Example 3
2
2
7
7
y xdy d xdx dx
( )dy dk f xdx dx
Sum and Difference RuleIf , then)()( xvxuy )()( xvxuy
Example 4
2
2
5
5
y x x
y x x
Example 53 2x x xy
x
Expectations Use pencils Use “=“ signs