VSLT (T&V)

TOPI
PAR
DIFFEREN

❑1.1 First Order Parti
❑1.2 Second Order Pa
❑1.3 Extremum of Fu

Variables
❑1.4 Lagrange Multip

IC 1 :
RTIAL
NTIATION

ial Derivatives 1
artial Derivatives
unctions of Two

plier

1.1 – First Order Pa

COURSE FRAMEWORK AND ST

a) Compute the first order
involving two variables x

PHD

dy =V du + U dv
dx dx dx

PHB du dv
dx dx
dy V −U

=
dx V 2

PLn

dy = f (x)
dx f (x)

artial Derivatives

TUDENT LEARNING TIME (SLT)

r partial derivatives
x and y

PExp

1) Salin semula soalan
2) Darab DIFF kuasa

PKuasa

1) Kuasa datang ke depan
2) Salin semula dalam kurungan
3) Kuasa berkurang satu
4) Darab DIFF dalam kurungan

2

1. Functions of several var
variables in a single fun

For example : a) f (x, y)

b) Z (x, y

2. The first order partial
by differentiating the
to a particular variabl
other variables fixed

riables consist of few
nction.

) = x2 + y2 − x − y

) = ln (x − y 2 )

l derivative is obtained
e function with respect
le, by keeping the
d.

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The First Order P

Description ‘ de

Partial derivative of f
x
z = f ( x, y)
f
with respect to x : y
Partial derivative of

z = f ( x, y)

with respect to y :

We know that f (
fx (x, y) = Z x and

Partial Derivative

Notations

el ’ operator f operator

z fx
x

z fy
y

(x, y) = Z then 4
d f y (x, y) = Z y

EXAMPLE 1:

Find fx and fy for f (x, y) =

Partial derivative of f
with respect to x

 fx ( x, y) = 4x + 3y

= 2x2 + 3x y − 6y2

Partial derivative of f
with respect to y

 fy ( x, y) = 3x −12 y 5

EXAMPLE 2:

Find f and f for each of th
x y

(a) f ( x, y) = 2x2 + 3y − 6y2

Ans : f = 4x , f = 3 −12 y
x y

he following :

(b) f ( x, y) = x2 y3 + x4 y

Ans : f = 2xy3 + 4x3 y , f = 3x2 y2 + x4 6
x y

EXAMPLE 3:

Find partial derivatives of f
and y

(a) f ( x, y) = ( x − 2y)3

(b) f ( x, y) = 4x2 + y2

(c) f ( )x, y = ex2y+2

(d ) f ( x, y) = ln ( x + 3y)2

( x, y) with respect to x

[Recall : The Power Rule]
[Recall : Square Root Functions]
[Recall : The Exponential Functions]

[Recall : Natural Logarithmic Functions]

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Find partial derivatives of

and y

(a) f ( x, y) = ( x − 2 y)3 [Re

f f ( x, y) with respect to x

ecall : The Power Rule]

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Find partial derivatives of

and y

(b) f ( x, y) = 4x2 + y2 [R

f f ( x, y) with respect to x

Recall : Square Root Functions]

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Find partial derivatives of

and y
(c) f ( )x, y = ex2y+2 [Recall : T

f f ( x, y) with respect to x

The Exponential Functions]

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Find partial derivatives of

and y

(d ) f ( x, y) = ln ( x + 3y)2 [

f f ( x, y) with respect to x

[Recall : Natural Logarithmic Functions]

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EXAMPLE 4:

Find fx for each of the follow

(a) f ( )x, y = x e2 2x2 +4 y

Ans :

( )( )a fx = e2x2 +4 y (b) ex−2 y 1 ln (3x)
4x3 + 2x , fx =  x +

[Using Product Rule : KAT+VU]

wing .

(b) f ( x, y) = ex−2y ln (3x)

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EXAMPLE 5:

Find f y for each of the follow

(a) f ( x, y) = 3x − 5y

2xy −1

Ans :

(a) fy = 5 − 6x2 , (b) fy = 3y − 2x − 3y ln y

(2xy −1)2 y (3y − 2x)2

[Using Quotient Rule : KAT+VU]

wing .

(b) f ( x, y) = ln y

3y − 2x

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EXAMPLE 6:

If z = ln x2 + y2, show that

x z + y z = 1
x y

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EXAMPLE 7: x − 2y
x2 − 2y2
Given that z = . Show th

1  z − z  = 1
+ 2y2  x y  x2 − 2y2 2
( )hatx2  

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1.2 – Second Order

COURSE FRAMEWORK AND ST

a) Compute the second or
involving two variables

Partial Derivatives

TUDENT LEARNING TIME (SLT)

rder partial derivatives
x and y

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❑ The second order partial derivativ
first order partial derivative with r
keeping the other fixed.

Description ‘ del

Partial derivative of fx  ( fx ) = 
with respect to x : x 
Partial derivative of f y
with respect to y :  ( f y ) = 
Partial derivative of fx x 
with respect to y :
Partial derivative of f y  ( fx ) = 
with respect to x : y 

 ( f y ) = 
x x

ve is obtained by differentiating the
respect to a particular variable, by

Notations

l ’ operator f operator

2 f 2z f xx
x2 x2 f yy
2 f 2z f xy
y 2 y 2
2z f yx
2 f yx
yx 2z 19
2 f xy
xy

EXAMPLE 8:

Given that f ( x, y) = 3x2 − 2xy

Find fxx, f yy, fxy and f yx,

y +5y2 .

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EXAMPLE 9:

Find 2z , 2z ,   z  for t
x2 y 2 y  x 

(a) z = x 2 y 3 + x 4 y

z xx = 2 y 3 + 12x 2 y, z xy = 6xy 2 + 4x3 , z yy = 6x 2 y

the following functions .

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