Partial Derivatives
Written by Dr. Julia Arnold
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 slide which follows 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))
In order to view the animation, you must have
the power point in slide show mode.
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  yx
Differentiate first with respect to x
and then with respect to y
  f   2 f
  
 f yx
y  y  xy
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
f ( x, y)  3xy2  2 y  5x2 y
Work the problem first then
check.
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
For comments on this presentation
you may email the author Dr. Julia
Arnold at jarnold@tcc.edu.