∂f

––

∂x

partial fractions 379

∂

∂f

––

∂y

∂f

––

∂y

∂f

––

∂x

∂f

––

∂y

∂f

––

∂x

y

one of two possible states: moving from Ato Bto C

could either have a clockwise or counterclockwise

sense. Every time a hockey player hits one puck

between the other two, the parity of the system changes.

This shows, for instance, that a hockey player will never

be able to return the three pucks to their original posi-

tions if she performs hits of this type 33 times.

As another example, it is impossible for 25 people

standing in a 5 ×5 grid of squares, one person per cell,

to shift places if each person is asked to move one place

over to a neighboring vertical or horizontal cell. A par-

ity argument explains why. If we imagine the cells col-

ored black and white according to a checkerboard

pattern, then the 13 people standing in black cells are

required to occupy just 12 white cells, rendering the

challenge unsolvable. (If, however, players are permitted

to take diagonal steps, then the puzzle can be solved.)

partial derivative Given a function f(x,y) of two

variables, the

DERIVATIVE

of fwith respect to just one

of the variables, treating the other variable as a con-

stant, is called a partial derivative of that function.

Specifically, the partial derivative of the function with

respect to x, denoted or sometimes fx, is the

LIMIT

:

and the partial derivative with respect to y, denoted

or fy, is:

Geometrically, the graph of the function z= f(x,y)

is a surface sitting in three-dimensional space, and the

quantity measures the slope of the graph surface

above the point (x,y) in the direction parallel to the

x-axis, and is the slope of the surface in the direction

parallel to the y-axis. Since one of the variables is being

treated as a constant, the partial derivative of a function

can be found by using the normal rules of differentia-

tion. For example, for the function f(x,y) = x2y3+ 2x+ y,

we have = 2xy3+ 2 and = 3x2y2+ 1.

The symbol ∂for “partial” was introduced in 1788

by the French mathematician J

OSEPH

-L

OUIS

L

AGRANGE

.

It resembles the dused for ordinary differentiation.

The nth partial derivative of a function with respect

to the same variable is denoted and differentiating

with respect to two different variables leads to mixed

partial derivatives: and the like.

The quantity , for example, is to be interpreted as

“differentiate first with respect to y, and then with

respect to x.” (That is, one performs the operations in

reverse order as indicated by the denominator.) For

instance, if f(x,y) = x2y3+ 2x+ ythen:

and

The equality

is no coincidence. Mathematicians have proved that the

order of partial differentiation does not matter as long

as all the partial derivatives involved are continuous

functions. The notion of a partial derivative extends to

functions of more than two variables.

A partial-differential equation is an equation that

contains partial derivatives of a function. For exam-

ple, the function f(x,y) = exy is a solution to the partial-

differential equation:

See also

CHAIN RULE

;

DIFFERENTIAL EQUATION

;

DIRECTIONAL DERIVATIVE

.

partial fractions Algebraic expressions containing

the ratio of two

POLYNOMIAL

s , with the degree of

p(x)

––

q(x)