7/28/2019 L5 Partial Derivatives
1/23
Chapter 2: Multivariable Calculus
Lecture 2: Partial Derivatives
byAssoc.Prof. Mai Duc
Thanh
7/28/2019 L5 Partial Derivatives
2/23
Rate of change of a function f(x,y)depends on the direction
Begin by measuring the rate of change if
we move parallel to thexoryaxes
These are called thepartial derivatives ofthe function
Partial Derivatives
7/28/2019 L5 Partial Derivatives
3/23
Definitions
Partial derivative with respect to x :
fx (x,y)
x
f(x,y) limh0
f(x h,y) f(x,y)
h
Partial derivative with respect to y :
fy (x,y)
yf(x,y) lim
h0
f(x,y h) f(x,y)
h
7/28/2019 L5 Partial Derivatives
4/23
f(x,y) = 4 - 2x2 - y2
Cut the surface withplanes:
x= 1 and y= -1
All meet at (1, -1, 1)
7/28/2019 L5 Partial Derivatives
5/23
The intersection of the graph off(x,y) with the planey = b is the graph of
g(x) = f(x,b)
Then:
Is the slope of the tangent line atx = a.
Partial with respect tox
fx
(a,b) d
dxf(x,b)
x a
g(a)
7/28/2019 L5 Partial Derivatives
6/23
g(x) = f(x,-1) = 3 - 2x2
g(x) = -4x
fx(1, -1) = g(1) = -4
- 1.0 - 0.5 0.5 1.0x
0.5
1.0
1.5
2.0
2.5
3.0
z
7/28/2019 L5 Partial Derivatives
7/23
The intersection of the graph off(x,y) with theplanex = a is the graph of
h(y) = f(a,y)
Then:
Is the slope of the tangent line at y = b.
Partial with respect to y
fy (a,b) d
dyf(a,y)
y b
h (b)
7/28/2019 L5 Partial Derivatives
8/23
h(y) = f(1, y) = 2 - y2
h(y) = -2y
fy(1, -1) = h(-1)
= 2
- 1.5 - 1.0 - 0.5 0.5 1.0 1.5
y
0.5
1.0
1.5
2.0
z
7/28/2019 L5 Partial Derivatives
9/23
The partial derivativesare the slopes of the
tangent lines parallelto thexz-plane andthe yz-plane.
(lines shown in red)
7/28/2019 L5 Partial Derivatives
10/23
To compute fx, treat yas a constant.
To compute fy, treatxas a constant.
Computing partial derivatives
7/28/2019 L5 Partial Derivatives
11/23
Find partial derivatives of the function
Solution.
To find fx, treat yas a constant. So
Similarly, to find fy, treatxas a constant. So
Example
2( , ) sin( )f x y xy x y
( , ) cos( ) 2xf x y y xy xy
2
( , ) cos( )yf x y x xy x
7/28/2019 L5 Partial Derivatives
12/23
Find partial derivatives of the function
Solution.
Note that fx, and fy are functions of two variables xand y
Example
2( , ) sin( )f x y xy x y
( , ) cos( ) 2 ( , )xf x y y xy xy g x y
2
( , ) cos( ) ( , )yf x y x xy x h x y
7/28/2019 L5 Partial Derivatives
13/23
The second derivative of a function of onevariable is very useful in determiningrelative maxima and minima
Second-order partial derivatives (partialderivatives of a partial derivative) are usedin a similar way for functions of two or
more variables
Higher Derivatives
7/28/2019 L5 Partial Derivatives
14/23
Second-order partial Derivatives
2
2
2
2
2
2
For a function ( , ) :
( , )
( , )
( , )
( , )
xx xx
yy yy
xy xy
yx yx
z f x y
z zf x y z
x x x
z zf x y z
y y y
z zf x y z
y x y x
z zf x y z
x y x y
7/28/2019 L5 Partial Derivatives
15/23
Example
f(x,y) 3x 2y 2sin(xy) y 3
fx (x,y) 6xy 2cos(xy)y
fy (x,y) 3x2 2cos(xy)x 3y 2
fxx(x,y) 6y 2sin(xy)y2
fxy(x,y)
y
fx (x,y) 6x 2sin(xy)xy 2cos(xy)
fyx(x,y)
xfy (x,y) 6x 2sin(xy)xy 2cos(xy)
fyy(x,y) 2sin(xy)x2
6y
7/28/2019 L5 Partial Derivatives
16/23
Clairauts Theorem
If (a,b) is in a disk D
and are continuous on D
then : ( , ) ( , )
xy yx
xy yx
f f
f a b f a b
7/28/2019 L5 Partial Derivatives
17/23
Functions of three variables
f(x,y,z) 3x 2yzz3y
x f(x,y,z) fx (x,y,z) 6xyz
y
f(x,y,z) fy (x,y,z) 3x2zz3
zf(x,y,z) fz(x,y,z) 3x
2y 3z2y
7/28/2019 L5 Partial Derivatives
18/23
Find all the first and the second partialderivatives of the function
Exercise
( , ) yf x y x
2
( , ) cos( )
x
yg x y t dt
7/28/2019 L5 Partial Derivatives
19/23
Recall
( ) ( )
x
a
df t dt f x
dx
7/28/2019 L5 Partial Derivatives
20/23
Example 1: The temperature of water at some point in ariver where a nuclear power plant discharges its hotwater is approximated by
x = temperature of river water before it reaches the plant
y= number of megawatts (in hundreds) of electricityproduced by the plant
a) Find and interprets
b) Find and interprets
Partial Derivatives as Rates ofchanges
( , ) 2 5 40T x y x y xy
(9,5)xT
(9,5)yT
7/28/2019 L5 Partial Derivatives
21/23
a) We have
Interpretation: This means that if x changes 1
degree from 9 to 9+1=10, then T approximatelychanges 7 degrees, while y remains constant=5
b)
Interpretation: This means that if y changes 1 unitfrom 5 to 5+1=6, then T approximately changes14 degrees, while x remains constant=9
Solution
( , ) 2 5 40
( , ) 2 , (9,5) 7x x
T x y x y xy
T x y y T
( , ) 5 , (9,5) 14y yT x y x T
7/28/2019 L5 Partial Derivatives
22/23
A company that manufactures computers hasdetermined that its production function isgiven by
x=size of labor force (work-hours/week)
y=amount of capital (units of $1000)
Find the marginal productivity of labor andcapital when x=50 and y=20, and interpretthe results
Example 2
4
2 3( , ) 500 800 3 4
yP x y x y x y x
7/28/2019 L5 Partial Derivatives
23/23
The marginal productivity of labor is given by
Interpretation: This means that if x change 1 unit from 50 to
51, then the production approximately changes 14000 units,while y remains constant at 20
The marginal productivity of capital is given by
Interpretation: This means that if y changes 1 unit from 20 to21, then the production approximately changes 300 units,while x remains constant at 50
Solution4
2 3( , ) 500 800 3
4
yP x y x y x y x
2
2
( , ) 500 6 3
(50,20) 500 6(50)(20) 3(50 ) 14000
x
x
P x y xy x
P
2 3
2 3
( , ) 800 3
(50,20) 800 3(50 ) 20 300
y
y
P x y x y
P