w5_stu differentiation 1

8/7/2019 w5_stu differentiation 1

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INTRODUCTION TOINTRODUCTION TODIFFERENTIATIONDIFFERENTIATIONINTRODUCTION TOINTRODUCTION TODIFFERENTIATIONDIFFERENTIATION


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DEFINITION

The derivative of a function f

is the function f whose value

at x is :

0

( ) ( )'( ) lim

h

f x h f xf x

hp

!

provided the limit exists.


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IMPORTANT IDEA

There are manyways to denote the

derivative of afunction.

You need to be

familiar with all ofthem.

'( )f x

dydx

22

dt

dt


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IMPORTANT IDEA

There are differentiation rules

that allow a shorter, easier way to

find derivatives. You will need to

memorize these rules.


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DIFFERENTIATION RULES

1.Constant Rule

[ ]d

c odx

!

The derivative of a constant function is 0.

There is no rate of change.


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EXAMPLE 1

?)3(.) !d

d

a

Find the derivative, if it exists :

Solution :


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Solution : step-by-step

b) Differentiate y = -2 with respect to x

Solution : Another way to write answer


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c) Find the derivative of

Solution :

2)( exf !d) Find the derivative of

Solution :


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Solution :

34ln)( !tsf) Find the derivative of

Solution :

e) Find the derivative of

where m is a constant

2)( mtf !


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2. Power Rule : (3 steps)


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EXAMPLE 2

Find the derivative, if it exists :

2

7

)()

))

mmfc

fbya

!

!!

3 2

5

2

)()

1)

)()

xxff

xxfe

xfd

!

!

!


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1xy !.a

Power Rule

Solution :


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7 xxf !.bPower Rule

Solution :


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2 mmf !.cSolution :

2 mxf !.dSolution :


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2

m

xf !.d


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5

1)

x

ye !

Solution :


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Solution : continue..


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5 3) xyf !

Solution : rewrite use

15

3

5

3 ! x

dx

dy

b

a

b a xx !Given: 5 3

xy!

Rewrite : 5

3

xy !


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Solution : continue..

5

2

5

3 ! x

5

2

5

3

x

!

5 25

3

x!

Change to power ofpositive

Change to root form


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3.Constant Multiple Rule

naxdx

d nxdx

da!

1. ! nnxa

Take out coefficient


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EXAMPLE 3 : Find the derivative of

36xy !

Solution : Step-by-step by using 1. ! nn nxaaxdx

d

dx

dy 36xdx

d! 36 x

dx

d!

21318)3(6 xx !!

Rewrite

Take out coefficientdifferentiate


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Solution : (simple way to write answer)

3

6xy!

!

dx

dy

6 )3(13

x2

18x!

Copy coefficient Differentiate3x

EXAMPLE 3 : Find the derivative of3

6xy !


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EXAMPLE 4 : Find the derivative of2

35

xy !

dxdy

!

23

5

xdxd 52

3 xdxd!

)5(2

3 15! x

2

154x

!

Solution : Step-by-step by using 1. ! nn nxaaxdx

d

Take out

coefficient

Rewrite

differentiate


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EXAMPLE 5: Find the derivative of5

2

3

x

y !

dx

dy

!

5

2

3

xdx

d

!

5

1

2

3

xdx

d

5

2

3

! x

dx

d

Rewrite

Rewrite

Solution : Step-by-step by using 1! nn axaxdx

d

Take out

coefficient


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52

3 ! xdx

d

Solution : continue.

1552

3 ! x 6

2

15 ! x

62

15

x!

Change to

power ofpositive

differentiate


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EXAMPLE6: Find the derivative of linearfunction y = 5x

Solution : By using short-cut

formula for linear function aaxdx

d!

Solution : Step-by step


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Example 7 : Find the derivative of

2

xy !

Solution : Step-by step


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Given :2

xy !

Solution : By using short-cut

formula for linear function aax

dx

d!


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4. Sum or Difference Rule

The derivative of the sum or difference is

the sum or difference of the derivatives.

? A!s)()( xgxfdx

d? A

)(xfdx

d? A)(xg

dx

ds


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Example 8 : Differentiate the following function

Solution : Cannot differentiate directly, so must rewrite

421

2

125 xxxy !

225

4

xxxy !

225

4xxxy !Given

rewrite


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Basic rules0)(.1 !C

dx

d

1)(.3 ! nn nxxdx

d

1)(.4! nn anxax

dxd

aaxdx

d

!)(.2!dxdy 144

21 x 121

21 x20

Given421

2

125 xxxy !

)4(2

1

2

1

2

321

xxdx

dy!

32

2

12 x

x!

Solution : continue.

differentiate


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DERIVATIVES OF THE

TRIGONOMETRIC

FUNCTIONS


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STANDARD FORM GENERAL FORM

xcosxsindxd

! )x('f)x(fcos)x(fsindxd

!

xsinxcosdx

d! )x('f)x(fsin)x(fcos

dx

d!

xsecxtandx

d 2! )x('f).x(fsec)x(ftandx

d 2!

xcotxcscxcscdx

d! )x('f).x(fcot)x(fcsc)x(fcsc

dx

d!

xtanxsecxsec

dx

d! )x('f).x(ftan)x(fsec)x(fsec

dx

d!

xcscxcotdx

d 2! )x('f).x(fcsc)x(fcot

dx

d 2!


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1)(cos

step

xf

1

)(sinstep

xf

1

2 )(sec

step

xf

2'

)(

step

xf

2

' )(step

xf

? A!)(sin xfdx

d

? A!)(cos xfdx

d

? A!)(tan xfdx

d 2

' )(

step

xf

General Form (2 steps)


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Example 9: Find the derivative of

xy 4sin!

!

dx

dy

x4cos4!

? A 2

'

1

)(.)(cos)(sin

stepstep

xfxfxfdx

d!

Use formula

xy 4sin!Solution :

1

4cos

step

x_2

)4(

step


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Example 10 : Find the derivative of

!

3cos6

xy

6!dx

dy

!

3

sin2x

? A 2

'

1

)(.)(sin)(cos

stepstep

xfxfxfdx

d!

Use formula

!

3cos6

xy

Solution :

Copy

coefficient

1

3sin

step

x

_ 2

3

1

step

Simplify

answer


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DERIVATIVES OF THE

EXPONENTIAL

FUNCTIONS


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STANDARDFORM

GENERAL FORM

xx

eedx

d! )('.

)()(

xfeedx

d xfxf!


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GENERAL FORM (2 steps)

_ 21)()( )('.

stepstep

xfxf xfeedx

d!

Example 11 :xey 3!

_ 13

step

xexe 33 !

_ 2

)3(

step

!

dx

dy

Copy exponential functionDifferentiate the power


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EXAMPLE 12 : Find the derivative ofxe

y52

1

!

Rewrite)52( xey !

xey 52!

152

step

xe xe 525 !

2

)50(

step

!

dx

dy

Copy exponential function

Differentiate the power

Solution : Given xey 52

1

!


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DERIVATIVES OF THE

LOGARITHMIC

FUNCTIONS


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STANDARDFORM

GENERAL FORM

xx

dx

d 1ln !

)(

)('ln

xf

xfxf

dx

d!


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GENERAL FORM (2 steps)

)()('ln

xfxfxf

dxd !

Example 13 : Differentiate )35ln( xy !

!

dx

dy

Step 1:Copy f(x)

Step 2 :Differentiate f(x)

Step 2 :Differentiate f(x)30

Step 1:Copy f(x))35( x

x35

3

!


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EXAMPLE 14 : Find the derivative of

x2coslny !

!dx

dy Step 2 :Differentiate f(x)2).2sin( x

Step 1:Copy f(x))2(cos xSolution :

!

x

x

2cos

2sin2

x2tan2!

By trigonometricidentity


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Conclusion : Differentiation Rules

0)( !Cdx

d1)( ! nn nxx

dx

d

1

)(

!nn

anxaxdx

daaxdx

d

!)(

1. Constant Rule

2. Linear function

3. Power Rule

4. Constant Multiple Rule


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? A 21

)('.)(cos)(sin

stepstep

xfxfxfdxd !

5. Trigonometric Function

? A 21

)('.)(sin)(cos

stepstep

xfxfxf

dx

d!

6.Exponential Function

? A_ 21

)()( )('.

stepstep

xfxf xfeedx

d!

? A 2

1

2 )('.)(sec)(tan

stepstep

xfxfxfdx

d!

7.Logarithmic Function

)(

)('ln

xf

xfxf

dx

d!

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w5_stu differentiation 1