12.3 Partial Derivatives.
Denition. If f is a function of two variables, its partial derivatives are the functions
fx and fy dened by
f (x + h, y) − f (x, y)
h→0
h
f (x, y + h) − f (x, y)
fy (x, y) = lim
h→0
h
fx (x, y) = lim
fx (x, y) represents the rate of change of the function f (x, y) as we change x and
hold y xed while fy (x, y) represents the rate of change of f (x, y) as we change y and
hold x xed.
Notation for partial derivatives: If z = f (x, y), we write
∂f
∂
∂z
=
f (x, y) =
= Dx f = D1 f = f1
∂x
∂x
∂x
∂f
∂
∂z
fy (x, y) =fy =
=
f (x, y) =
= Dy f = D2 f = f2
∂y
∂y
∂y
fx (x, y) =fx =
Rule for nding partial derivatives of z = f (x, y):
1. To nd fx , regards y as a constant and dierentiate f (x, y) with respect to x.
2. To nd fy , regards x as a constant and dierentiate f (x, y) with respect to y .
Example 1. Find the rst partial derivatives of the following functions:
(a) f (x, y) = x3 − x2 y + y 3
(b) f (x, y) = x3 sin(xy)
(c) f (x, y) = ex
Example 2. If f (x, y) = x3 + y 5 ex , nd fx (0, 1) and fy (0, 1).
Example 3. Find ∂z/∂x and ∂z/∂y if z is dened implicitly as a function of x and y
by the equation
x3 + y 3 + z 3 + 6xyz = 1
Example 4. The temperature T at a point (x, y) on a at metal plate is given by
T (x, y) =
80
,
1 + x2 + y 2
where T is measured in ◦ C and x, y in meters. Find the rate of change of temperature
with respect to distance at the point (1, 2) in the y -direction.
Functions of more that two variables
Partial derivatives can also be dened for functions of three or more variables. For
example, if f is a function of three variables x, y , and z , then its partial derivative
with respect to x is dened as
f (x + h, y, z) − f (x, y, z)
h→0
h
fx (x, y, z) = lim
and it is found by regarding y and z as constants and dierentiating with respect to
x.
In general, if u is a function of n variables, u = f (x1 , x2 , . . . , xn ), its partial derivative with respect to the i-th variable xi is
∂u
f (x1 , . . . , xi−1 , xi + h, xi+1 , . . . , xn ) − f (x1 , . . . , xi , . . . , xn )
= lim
∂xi h→0
h
Example 5. Find fx , fy and fz if f (x, y, z) = yexyz .
Higher Derivatives
If z = f (x, y) is a function of two variables, then its rst order partial derivatives fx
and fy are also functions of x and y , we can dierentiate fx and fy with respect to x
or y . Then (fx )x , (fx )y , (fy )x and (fy )y are called the second partial derivatives
of f . We use the following notation:
(fx )x
(fx )y
(fy )x
(fy )y
∂ 2f
∂ 2z
∂ ∂f
=
=
=fxx = f11 =
∂x ∂x
∂x2
∂x2
∂ ∂f
∂ 2f
∂ 2z
=fxy = f12 =
=
=
∂y ∂x
∂x∂y
∂x∂y
∂ ∂f
∂ 2f
∂ 2z
=fyx = f21 =
=
=
∂x ∂y
∂y∂x ∂y∂x
∂ 2f
∂ 2z
∂ ∂f
= 2 = 2
=fyy = f22 =
∂y ∂y
∂y
∂y
Clairaut's Theorem. Suppose f is dened on a disk D that contains the point (a, b).
If the functions fxy and fyx are both continuous on D, then
fxy (a, b) = fyx (a, b)
Example 6. Find all the second partial derivatives for the function z = ex sin y .
Example 7. Show that the function u = cos(x − ct) is a solution of the wave equation
utt = c2 uxx
Partial derivative of order three or higher can also be dened. For instance,
fxyy
∂
= (fxy )y =
∂y
∂ 2f
∂x∂y
=
∂ 3f
∂y 2 ∂x
Example 8. Find fxyz for the function f (x, y, z) = cos(xy + z).