2 1. ADVANCED DERIVATIVES
Where before we had points .x; y/, we now have points .x; y; z/. We’ve looked at the graph of
functions like y D x
2
over and over. Now let’s look at the three-dimensional analog:
z D x
2
C y
2
for 4 x; y 4. A graph of the function is shown in Figure 1.2.
-4
-3
-2
-1
0
1
2
3
4
-4
-3
-2
-1
0
1
2
3
4
0
5
10
15
20
25
30
35
f(x,y)=sqr(x)+sqr(y):
Figure 1.2: A graph of f .x; y/ D x
2
C y
2
for 4 x; y 4.
1.1 PARTIAL DERIVATIVES
So far in this text we have had one independent and one dependent variable. Now we have two
independent variables, which has a huge effect on derivatives. e number of directions on the
number line is two – left and right or plus and minus. As we learned in Fast Start Integral Calculus,
there are an infinite number of directions when there are two variables to choose directions
among. Each unit vector starting at the origin points in a different direction. Every single vector
can be written as:
Ev D .a; b/ D a .1; 0/ C b .0; 1/
1.1. PARTIAL DERIVATIVES 3
In other words, each vector is a combination of the two fundamental vectors Ee
1
D .1; 0/ and
Ee
2
D .0; 1/.
On the surface of a function z D f .x; y/, the function has a rate of change in every direction at
every point. Pick a direction – the slope in that direction is the rate of change in that direction.
e basis for all of these rates of change are the partial derivatives – derivatives in the direction
of x and y.
Knowledge Box 1.1
Partial Derivatives
If z D f .x; y/ is a function of two variables, then there are two fundamental
derivatives
z
x
D
@f
@x
and z
y
D
@f
@y
ese are called the partial derivatives of z (or f ) with respect to x and y,
respectively.
In order to take the partial derivative of a function with respect to one variable, all other variables
are treated as constants. is is most easily understood through examples.
Example 1.1 Suppose that z D x
2
C 3xy Cy
2
. Find z
x
and z
y
.
Solution:
z
x
D 2x C 3y and z
y
D 3x C 2y
To see this, notice that, when we are taking the derivative with respect to x, the derivative of
3xy is 3y, because y is treated as a constant. Similarly, the derivative of 3xy with respect to y
is 3x. e derivatives for x
2
and y
2
are 2x and 2y when they are the active variable and zero
when the other variable is active, because the derivative of a constant is zero.
˙
4 1. ADVANCED DERIVATIVES
It is important to remember that every derivative rule we have learned so far applies when we are
taking partial derivatives – the product, quotient, and chain rules and all the individual formulas
for functions.
Example 1.2 Find
@f
@x
if
f .x; y/ D
x
x
2
C y
2
Solution:
is problem requires the quotient rule. Since a “prime” hash-mark doesn’t carry its identity
(with respect to x or with respect to y) we used the symbol
@
@x
to mean “derivative with respect
to x.”
is means that
@
@x
x
x
2
C y
2
D
@
@x
x .x
2
C y
2
/
@
@x
.x
2
C y
2
/ x
.
x
2
C y
2
/
2
D
1 .x
2
C y
2
/ 2x x
.
x
2
C y
2
/
2
D
y
2
x
2
.
x
2
C y
2
/
2
˙
e next example uses the chain rule. It also uses the additional notation “f
y
” as another way of
saying “the partial derivative of f .x; y/ with respect to y.”
Example 1.3 Find f
y
.x; y/ if
f .x; y/ D sin.2xy C 1/
Solution:
f
y
.x; y/ D cos.2xy C 1/ 2x
˙
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