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Total differential of a function z = z(x,y)
From the last equation, if we multiply by dt, we get the total differential of the
function z = z(x,y), i.e., dz =
A different version of the chain rule applies to the case in which z = f(x,y), x
= x(u,v), y = y(u,v), so that z = f[x(u,v), y(u,v)].
represent the chain rule for this situation:
z
z
u
x
Determining extrema in functions of two variables
In order for the function z = f(x,y) to have an extreme point (extrema) at (x
its derivatives ∂f/∂x and ∂f/∂y must vanish at that point. These are necessary
conditions. The sufficient conditions for the function to have an extreme at
) are ∂f/∂x = 0, ∂f/∂y = 0, and ∆ = (∂
point (x
,y
o
o
) is a relative maximum if ∂
> 0. The point (x
,y
o
o
2
2
minimum if ∂
> 0. The value ∆ is referred to as the discriminant.
f/∂x
⋅
2
2
2
If ∆ = (∂
f/∂x
)
(∂
f/∂y
saddle point, where the function would attain a maximum in x if we were to
hold y constant, while, at the same time, attaining a minimum if we were to
hold x constant, or vice versa.
Example 1 – Determine the extreme points (if any) of the function f(X,Y) = X
2
3X-Y
+5. First, we define the function f(X,Y), and its derivatives fX(X,Y) =
∂f/∂X, fY(X,Y) = ∂f/∂Y. Then, we solve the equations fX(X,Y) = 0 and fY(X,Y)
= 0, simultaneously:
∂z/∂x)
(
⋅
dx + (∂z/∂y)
x
z
y
z
,
u
y
u
v
2
2
2
)-[∂
f/∂x∂y]
< 0, we have a condition known as a
⋅
dy.
The following formulas
z
x
z
y
x
v
y
v
⋅
2
2
2
2
2
f/∂x
)
(∂
f/∂y
)-[∂
f/∂x∂y]
2
2
f/∂x
< 0, or a relative
Page 14-5
,y
),
o
o
2
3
-
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