mat 1221 survey of calculus section 2.2 some rules for differentiation

MAT 1221Survey of Calculus

Section 2.2 Some Rules for Differentiation

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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