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25.3 Partial derivatives 825
D
C
A
z = z(x,y)
x
z
y
E
B
Figure25.1
Theheight ofthe surface
above thex--yplane isz.
move away from A the value of z decreases. A minimum point is similarly defined.
At a minimum point, the value ofzis smaller than thezvalue at nearby points. This is
illustratedbypointB.PointCillustratesasaddlepoint.Atasaddlepoint,zincreasesin
one direction, axis D on the figure, but decreases in the direction of axis E. These axes
are at right angles to each other. The term ‘saddle’ is descriptive as a horse saddle has
a similar shape. Maximum points, minimum points and saddle points are considered in
greater depth inSection 25.7.
25.3 PARTIALDERIVATIVES
Consider
z =z(x,y)
thatis,zisafunctionoftheindependent variablesxandy.Wecandifferentiatezeither
w.r.t.x,orw.r.t.y.Weneedsymbolstodistinguishbetweenthesetwocases.Whenfinding
the derivative w.r.t. x, the other independent variable, y, is held constant and only
x changes. Similarly when differentiating w.r.t. y, the variable x is held constant. We
write ∂z
∂x todenotedifferentiationofzw.r.t.xforaconstanty.Itiscalledthefirstpartial
derivative of z w.r.t. x. Similarly, the first partial derivative of z w.r.t. y is denoted ∂z
∂y .
Referring to the surfacez(x,y), ∂z gives the rate of change ofzmoving only in thex
∂x
direction, and henceyisheld fixed.
Ifz =z(x,y), thenthe firstpartialderivatives ofzare
∂z
∂x
and
∂z
∂y
If we wish to evaluate a partial derivative, say ∂z
∂x , at a particular point (x 0 ,y 0
), we
indicate thisby
∂z
∂x (x 0 ,y 0 ) or ∂z
∣
∂x
∣
(x0 ,y 0
)
justas wedid forfunctions of one variable.