3Partial differentiation
IA Differential Equations
3.1 Partial differentiation
So far, we have only considered functions of one variable. If we have a function
of multiple variables, say
f
(
x, y
), we can either differentiate it with respect to
x
or with respect to y.
Definition (Partial derivative). Given a function of several variables
f
(
x, y
),
the partial derivative of
f
with respect to
x
is the rate of change of
f
as
x
varies,
keeping y constant. It is given by
∂f
∂x
y
= lim
δx→0
f(x + δx, y) −f(x, y)
δx
Example. Consider
f
(
x, y
) =
x
2
+
y
3
+
e
xy
2
. Computing the partial derivative
is equivalent to computing the regular derivative with the other variables treated
as constants. e.g.
∂f
∂x
y
= 2x + y
2
e
xy
2
.
Second and mixed partial derivatives can also be computed:
∂
2
f
∂x
2
= 2 + y
4
e
xy
2
∂
2
f
∂y∂x
=
∂
∂y
∂f
∂x
= 2ye
xy
2
+ 2xy
3
e
xy
2
It is often cumbersome to write out the variables that are kept constant. If
all other variables are being held constant, we simply don’t write them out, and
just say
∂f
∂x
.
Another convenient notation is
Notation.
f
x
=
∂f
∂x
, f
xy
=
∂
2
f
∂y∂x
.
It is important to know how the order works in the second case. The left
hand side should be interpreted as (
f
x
)
y
. So we first differentiate with respect
to x, and then y. However, in most cases, this is not important, since we have
Theorem. If f has continuous second partial derivatives, then f
xy
= f
yx
.
We will not prove this statement and just assume it to be true (since this is
an applied course).