SOC Notes 6.2, O’Brien, F12
Calc & Its Apps, 10th ed, Bittinger
6.2
Partial Derivatives
I.
Review of Differentiation for Functions of One Variable
A.
The Definition of the Derivative
f x  h  f x 
,
h
h0
For the function f, the derivative of f at x is defined as f x   lim
provided the limit exists.
B.
Summary of Formulas for Differentiation
Assuming c and n stand for constants, f and g stand for differentiable functions of x, and g ≠ 0:
C.
II.
The Constant Rule
d
c0
dx
The Simple Power Rule
d n
x  n  x n1
dx
[special case:
d
x  1]
dx
The Constant Multiple Rule
d
c  f   c  f 
dx
[special case:
d
c  x   c ]
dx
The Sum & Difference Rule
d
f  g  f   g
dx
The Product Rule
d
f  g  f   g  g  f
dx
The Quotient Rule
d
dx
The Generalized Power Rule
d
gn  n  gn1  g
dx
The Chain Rule
d
f  g  d f g  f g g
dx
dx
The Derivative of ln [f(x)]
f x 
d
ln f x  
dx
f x 
[special case:
d
1
ln x  ]
dx
x
The Derivative of e f x 
d f x 
e
 e f x   f x 
dx
[special case:
d x
e  ex ]
dx
 f  f   g  g  f
  
g2
g
Interpretations of the Derivative
1.
Slope of the line tangent to f(x) at the point x1, f x1 
2.
Instantaneous rate of change
3.
Marginal cost, marginal revenue, marginal profit
4.
Velocity and acceleration
Partial Derivatives – Formal Definitions
Functions of several variables have several derivatives, one for each variable. Each of these is called a
partial derivative.
The partial derivative of z = f(x, y) with respect to x is
f x  h, y   f x, y 
z
z
 lim
. In
, x is increased
x h0
h
x
by h, but y is held constant.
1
SOC Notes 6.2, O’Brien, F12
Calc & Its Apps, 10th ed, Bittinger
Note:
To distinguish between a regular derivative and a partial derivative with respect to a specified
variable, we use the lowercase Greek letter delta   rather than d.
The partial derivative of z = f(x, y) with respect to y is
f x, y  h  f x, y 
z
z
 lim
. In
, y is increased
y h0
h
y
by h, but x is held constant.
Note:
III.
Since we can use the formulas for differentiation reviewed in section IB, we will rarely need to use
the formal definition to find a partial derivative.
Partial Derivatives – Informal Definitions
The partial derivative of z = f(x, y) with respect to x is obtained by differentiating z = f(x, y) with respect
f z
to x, treating y as a constant. The notation for this partial derivative is
,
, or f x .
x x
The partial derivative of z = f(x, y) with respect to y is obtained by differentiating z = f(x, y) with respect
f z
to y, treating x as a constant. The notation for this partial derivative is
,
, or f y .
y y
Example 1:
For z  f x, y  x 3 y 4  5xy  4y 2 , find
z
z
and
.
x
y
z
, we differentiate with respect to x, treating y as a constant.
x
To find
z
 3x 2 y 4  5y
x
z
, we differentiate with respect to y, treating x as a constant.
y
To find
z
 4x 3 y 3  5x  8y
y
Example 2:
Given f x, y  y  ln x  2y  , find f x and f y .
fx  y 
Example 3:
y
1

x  2y x  2y
Given f x, y  
f x, y  
Example 4:
2y
2
 y  ln x  2y  
x  2y
x  2y
x y

, find f x and f y .
y 5x
y
1
x  x 1
y
5
f x, y   xy 1 
f y  1 ln x  2y  
1
y
5x
fx 


y
1 y
1
  1 x 2  
y 5
y 5x 2


f y  x  1 y 2 
1
x
1


2 5x
5x
y
Given f x, y,    5xy   2x  y  8 , find f x , f y , and f  .
f x  5y   2  5y  2
f y  5x   1  5x  
f   2x  y  8
2
SOC Notes 6.2, O’Brien, F12
Calc & Its Apps, 10th ed, Bittinger
IV.
Evaluating a Partial Derivative
To evaluate a partial derivative of z = f(x, y) at a point (a, b), first differentiate and then evaluate.
f
z
f
z
Such problems may be written
,
, f x a, b ,
,
, or f y a, b .
x a, b  x a, b 
y a, b  y a, b 
Example 5:
z
 3x
y
z
 6x 2  3y  1
x
Example 6:
z
 6(2) 2  3 3  1  14
x 2, 3 
z
 30  0
y 0, 5 
Given f x, y   x 2  y 2 , find f x , f y , f x  2, 1 , and f y  3,  2 .

f x, y   x 2  y 2  x 2  y 2
V.
z z z
z
,
,
, and
.
x y x 2, 3 
y 0, 5 
Given z  2x 3  3xy  x , find




fx 
1
1 2
x  y 2 2  2x 
2
fy 
1
1 2
x  y 2 2  2y 
2

1
2
x
x2  y2
y
x2  y2
f x  2, 1 
f y  3,  2 
2
 22  12

2
 32   22
2

2 5
5
2

5

13
2 13
13
Interpreting Partial Derivatives Geometrically
A function z = f(x, y) represents a surface in three-dimensional space and its partial derivatives give the
slope or steepness along the surface of the function in different directions. For a surface z = f(x, y) and a
point x 0 , y 0 , z 0  on this surface, the partial derivative with respect to x gives the slope of the tangent
line at x 0 , y 0 , z 0  in the positive x-direction. Similarly, the partial derivative of f with respect to y gives
the slope of the tangent line at x 0 , y 0 , z 0  in the positive y-direction.
VI.
Second-Order Partial Derivatives
The second-order partial derivatives of z = f(x, y), a function in two variables, are:
1.
f xx or
2f
x 2
Find f x , then differentiate that result with respect to x. Hold y constant each
time.
2.
f xy or
2f
yx
Find f x , holding y constant. Then differentiate that result with respect to y,
holding x constant.
3.
f yx or
2f
xy
Find f y , holding x constant. Then differentiate that result with respect to x,
holding y constant.
4.
f yy or
2f
y 2
Find f y , then differentiate that result with respect to y. Hold x constant each
time.
Note:
Often (but not always), f xy = f yx .
3
SOC Notes 6.2, O’Brien, F12
Calc & Its Apps, 10th ed, Bittinger
Caution:
For second order partial derivatives in delta notation,  2 does not mean
“delta squared”. Also, the order of differentiation is the opposite of the way it is
2f
indicates we should find f y first and then differentiate that result with
xy
respect to x.
written.
Example 7:
Given f x, y  x 4 y 3  x 2 y 3 , find the four second order partial derivatives.
f x  4x 3 y 3  2xy 3
f xy  12x 3 y 2  6xy 2
f xx  12 x 2 y 3  2y 3
f y  3x 4 y 2  3x 2 y 2
f yx  12x 3 y 2  6xy 2
VII.
f yy  6x 4 y  6x 2 y
An Economics Application: The Cobbs-Douglas Production Function
One model of production that is frequently considered in business and economics is the
Cobb-Douglas production function:
px, y   Ax a y 1a , for A > 0 and 0 < a < 1
where p is the number of units produced with x units of labor and y units of capital.
(Capital is the cost of machinery, buildings, tools, and other supplies.) A and a depend on
the means of production and vary between companies and products.
p
The partial derivative
is the marginal productivity of labor. It gives the rate of increase of
x
production with respect to units of labor.
p
The partial derivative
is the marginal productivity of capital. It gives the rate of increase of
y
production with respect to units of capital.
A Cobb-Douglas production function is consistent with the law of diminishing returns.
That is, if one input (either labor or capital) is held fixed while the other increases infinitely,
then production will eventually increase at a decreasing rate.
Example 8:
The Cobb-Douglas model
2
1
Bombastics has the following production function for a certain product: px, y  225 x 3 y 3
where p is the number of units produced with x units of labor and y units of capital.
a.
Find the number of units produced with 27 units of labor and 125 units of capital.
b.
Find the marginal productivities.
c.
Evaluate the marginal productivities at x = 27 and y = 125.
a.
p27, 125   225 27  3 125  3  10,125
b.
 2 1  1
1 1
p
 225  x 3  y 3  150 x 3 y 3
x
3

c.
2
1
2 
p
1 2
 225 x 3  y 3
y
3
2

2
  75 x 3 y 3


1
1
p
27, 125   150 27  3 125  3  250 Production increases by 250 for each additional unit of labor
x
2
2
p
27, 125   75 27  3 125  3  27 Production increases by 27 for each additional unit of capital
x
4