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HUT – DEPARTMENT OF MATH. APPLIED--------------------------------------------------------------------------------------------------------

GIAÛI TÍCH 2 (QUOÁC TEÁ)

LECTURE 01:FUNCTIONS OF SEVERAL VARIABLES:

PARTIAL DERIVATIVES

STEWART: PAGES 855 878

   PhD. NGUYEÃN QUOÁC LAÂN (August, 2015)

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LECTURE 1 CONTENTS---------------------------------------------------------------------------------------------------------------------------------

1- FUNCTION OF 2 VARIABLES: DEFINITION – GRAPH.

2- PARTIAL DERIVATIVES

4- INCREMENTAL FORMULA - DIFFERENTIAL

3- IMPLICIT DIFFERENTIATION

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FUNCTIONS OF TWO VARIABLES--------------------------------------------------------------------------------------------------------------------------------------

( ) ( )

( ) ( ){ }   f  R y x f  y x D

 D y x y x f  z

 of domaintheisdefined is,|,setThe

.,,,formulaaisvariablestwoof f functionA

2⊂=

∈=

( )

( ) ( ) D.domain&1,2,2,1find 

,,For:Example

 f  f 

 x y y x f    −=

But the graph of z = f(x, y) is a

surfacecurveaisf(x)yof graphThe   =

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EXAMPLE--------------------------------------------------------------------------------------------------------------------------------------

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EXAMPLE--------------------------------------------------------------------------------------------------------------------------------------

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PARTIAL DERIVATIVES OF z = f(x, y)--------------------------------------------------------------------------------------------------------------------------------------

( )

⎪⎪⎭

⎪⎪⎬

⎫

⎢

⎢⎢⎢

⎣

⎡

==∂

∂==

==∂

∂==

=

 f  D f  D y

 f 

 f  f 

 f  D f  D x

 f  f  f 

 y x f  z

 y y y

 x x x

2

/

1/

 :,: Notation

Rule for finding partial derivatives of z = f(x, y)

( ) y x orto

respectwith

sDerivative

Partial

1/ To find f x, regard y as a constant and differentiate z = f(x, y)with respect to x

2/ To find f y, regard x as a constant and differentiate z = f(x, y)

with respect to y

⎢⎣

⎡

=

=⇒−+=

?

?2:1Example 2323

 y

 x

 f 

 f  y y x x f  ⎢

⎣

⎡

=

=⇒=

?

?:2Exam.

 y

 x y

 f 

 f  x f 

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HIGHER – ORDER PARTIAL DERIVATIVES--------------------------------------------------------------------------------------------------------------------------------------

( )( ) ( )

( ) ( )

K

,43defineweSimilarly,.of sderivative partialorder-Second 

:,

,,of Functions,

thrd 

 f 

 z z z z

 z z z z y x

 z

 z y x f  z

 yy y y yx x y

 xy y x xx x x

 y

 x

⎢⎢⎣

⎡

==

==⇒

⎭⎬⎫

⎢⎣

⎡⇒=

K2

32

2

2

  partial,second mixed :,:Symbol y x

 f  f 

 y x

 f  f 

 x

 f  f   xyy xy xx

∂∂

∂=

∂∂

∂=

∂

∂=

2323 2of sderivative partialsecond theFind :Example   y y x x f    −+=

 therefore43,23haveWe:Answer  2232  y y x f  xy x f   y x   −=+=

46 ,6 ,6,26 22

2

2

2

2

2

32

2

−=∂

∂=∂∂

∂=∂∂

∂+=∂

∂ xy

 y z xy

 x y z xy

 y x z y x

 x z

...: partialsmixed ithEquality w2

3

2

33

 x y

 f 

 y x

 f 

 x y x

 f  f  f   yx xy

∂∂

∂=

∂∂

∂=

∂∂∂

∂⇒=

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IMPLICIT PARTIAL DIFFERENTIATION--------------------------------------------------------------------------------------------------------------------------------------

( ) ( ) ?,0,, byimplicitlydefined is,Function =⇒==   y x   z z z y xF  y x z z

( ) 16 bydefined is,if ,Find :Example 333 =+++=   xyz z y x y x z z z z  y x

:constantaasgconsiderin,to

respectwithimplicitlyequationthe

atedifferentiwe,find To:Answer 

 y x

 z x

( )106633 22 =+++   x x   xyz yz z z x

 xy z

 yz x z z  x x

2

2:for(1)Solving

2

2

+

+−=

 xy z

 xz y z z y

 z

 y y

 y

2

2forsolveand to

respectateDifferenti:forSimilarly

2

2

+

+−=⇒

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APPROXIMATION: INCREMENTAL FORMULA--------------------------------------------------------------------------------------------------------------------------------------

At some factory, the daily output is

units, where K – capital measured in units of $1000 and L –

labor force measured in worked – hours. The current capital is

$900.000 and 1000 worker – hours of labor are used each day.

Estimate the change in output that will result if capital isincreased by $1000 and labor is increased by 2 worker – hour

)Douglas!(Cobb60 31

2

1

 LK Q =

( ) ( ) ( ) ( ) ( )   yba f  xba f ba f  yb xa f  z y x f  z  y x   Δ⋅+Δ⋅≈−Δ+Δ+=Δ= ,,,,:,zinchangeThe

4 4 4 4 34 4 4 4 21

units222030:Ans.32213121

=Δ+Δ=Δ∂

∂

+Δ∂

∂

≈Δ

  −−

 L LK K  LK  L L

Q

K K 

Q

Q

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DIFFERENTIAL---------------------------------------------------------------------------------------------------------------------------

Given z = f(x, y), x & y: independent variables Differential

)number!realas,(Consider dydxdy

 y

 f dx

 x

 f dy f dx f dz  y x

∂

∂+

∂

∂=+=⇒

( )2,1 b/ a/ find let2 22 dzdz y x z   +=Given:Example

( )   dydxdz y

 x

 y x

 ydy xdxdy zdx zdz

 y x

 y z

 y x

 x z

 y x

 y x

3

4

3

12,1

2

1 b/ 

2

2a/

2

2 ,

2

22

2222

+=⇒⎩⎨⎧

=

=

+

+=+=

⇒

+

=

+

= sderivativePartial:Answer

Second order differential d2z

222 2   dy zdxdy zdx z zd   yy xy xx   ++=( )1,2find 

2Given:Example

2

2323

 zd 

 y y x x f    −+=

222

82414:Answer    dydxdydx zd    ++=