Chapter 7: The Cross Product of Two Vectors In Space
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Transcript Chapter 7: The Cross Product of Two Vectors In Space
Chapter 8: Partial
Derivatives
Section 8.3
Written by Dr. Julia Arnold
Associate Professor of Mathematics
Tidewater Community College, Norfolk Campus, Norfolk, VA
With Assistance from a VCCS LearningWare Grant
In this lesson you will learn
•about partial derivatives of a function of two variables
•about partial derivatives of a function of three or more variables
•higher-order partial derivative
Partial derivatives are defined as derivatives of a
function of multiple variables when all but the variable of
interest are held fixed during the differentiation.
Definition of Partial Derivatives of a Function of Two Variables
If z = f(x,y), the the first partial derivatives of f with respect to x
and y are the functions fx and fy defined by
f x x, y
x lim0
f y x, y
y lim0
Provided the limits exist.
f x x, y f ( x, y )
x
f x, y y f ( x, y )
y
To find the partial derivatives, hold one variable constant and
differentiate with respect to the other.
Example 1: Find the partial derivatives fx and fy for the function
f ( x, y) 5x4 x2 y 2 2x3 y
To find the partial derivatives, hold one variable constant and
differentiate with respect to the other.
Example 1: Find the partial derivatives fx and fy for the function
f ( x, y) 5x4 x2 y 2 2x3 y
Solution:
f ( x, y) 5x 4 x 2 y 2 2 x3 y
f x ( x, y) 20 x3 2 y 2 x 6 yx 2
f y ( x, y) 2 x 2 y 2 x3
Notation for First Partial Derivative
For z = f(x,y), the partial derivatives fx and fy are denoted
by
z
f ( x , y ) f x x, y z x
x
x
and
z
f ( x, y ) f y x, y z y
y
y
The first partials evaluated at the point (a,b) are denoted by
z
x
( a ,b )
f x a, b and
z
y
( a ,b )
f y a, b
Example 2: Find the partials fx and fy and evaluate them at the
indicated point for the function
f ( x, y )
xy
at (2, 2)
x y
Example 2: Find the partials fx and fy and evaluate them at the
indicated point for the function
f ( x, y )
Solution:
f ( x, y )
xy
at (2, 2)
x y
xy
at (2, 2)
x y
f x x, y
x y y xy
( x y)
f x 2, 2
f y x, y
2
2
2
(2 2 ) 2
x y x xy
f y 2, 2
( x y)
x2
( x y)2
2
xy y 2 xy
( x y)
2
y2
( x y)2
4 1
16
4
x 2 xy xy
( x y)
4 1
16 4
2
x2
( x y)2
The following slide shows the geometric interpretation of the partial
derivative.
For a fixed x, z = f(x0,y) represents the curve formed by intersecting
the surface z = f(x,y) with the plane x = x0.
f x x0 , y0 represents the slope of this curve at the point (x0,y0,f(x0,y0))
Thanks to http://astro.temple.edu/~dhill001/partial-demo/
For the animation.
Definition of Partial Derivatives of a Function of Three or More
Variables
If w = f(x,y,z), then there are three partial derivatives each of
which is formed by holding two of the variables
w
f x x, y , z
x
x lim 0
w
f y x, y , z
y
y lim 0
w
f z x, y , z
z
z lim 0
f x x, y, z f ( x, y, z )
x
f x, y y , z f ( x, y , z )
y
f x, y, z z f ( x, y, z )
z
In general, if
w f ( x1 , x2 ,...xn ) there are n partial derivatives
w
f xk x1 , x2 ,...xn , k 1, 2,...n
xk
where all but the kth variable is
held constant
Notation for Higher Order Partial Derivatives
Below are the different 2nd order partial derivatives:
f 2 f
2 f xx
x x x
Differentiate twice with respect to x
f 2 f
2 f yy
y y y
Differentiate twice with respect to y
f 2 f
f xy
y x yx
Differentiate first with respect to x
and then with respect to y
f 2 f
f yx
y y xy
Differentiate first with respect to
y and then with respect to x
Theorem
If f is a function of x and y such that fxy and fyx are
continuous on an open disk R, then, for every (x,y) in R,
fxy(x,y)= fyx(x,y)
Example 3:
Find all of the second partial derivatives of
Work the problem first then
check.
f ( x, y) 3xy2 2 y 5x2 y
Example 3:
Find all of the second partial derivatives of
f ( x, y) 3xy2 2 y 5x2 y
f ( x, y ) 3 xy2 2 y 5 x 2 y
f x ( x, y) 3 y 2 10 xy
f xx ( x, y) 10 y
f ( x, y ) 3 xy2 2 y 5 x 2 y
f y ( x, y) 6 xy 2 5 x 2
f yy ( x, y) 6 x
f ( x, y ) 3 xy2 2 y 5 x 2 y
f x ( x, y) 3 y 2 10 xy
f xy ( x, y) 6 y 10 x
f ( x, y ) 3 xy2 2 y 5 x 2 y
f y ( x, y) 6 xy 2 5 x 2
f yx ( x, y) 6 y 10 x
Notice that fxy = fyx
Example 4: Find the following partial derivatives for the
function
x
f ( x, y, z) ye xln z
a.
f xz
b.
f zx
c.
f xzz
d.
f zxz
e.
f zzx
Work it out then go to the next slide.
Example 4: Find the following partial derivatives for the
function
x
f ( x, y, z) ye xln z
a.
f xz
f ( x, y, z) yex x ln z
f x ( x, y, z) yex ln z
1
f xz ( x, y, z)
z
b.
f zx
f ( x, y, z ) ye x x ln z
x
z
1
f zx ( x, y, z )
z
f z ( x, y, z )
Again, notice that the 2nd
partials fxz = fzx
c.
f xzz
f ( x, y, z) ye x x ln z
f x ( x, y, z) ye x ln z
1
z
1
f xzz ( x, y, z) 2
z
f xz ( x, y, z)
e.
f zxz
f ( x, y, z) ye x x ln z
x
z
1
f zx ( x, y, z)
z
1
f zxz ( x, y, z) 2
z
f z ( x, y, z)
f ( x, y, z) ye x x ln z
x
z
x
f zz ( x, y, z) 2
z
1
f zzx ( x, y, z) 2
z
f z ( x, y, z)
Notice
All
Are Equal
d.
f zzx
Go to BB for your exercises.