
Differentiation 4-51
Proof: Since Z is a homogeneous function of x and y of degree n, it can be expressed as
Z x f
y
x
Z
x
x f '
y
x
y
x
nx
n
n n
=
⇒
∂
∂
=
−
+
.
2
−−
1
f
y
x
∂
∂
=
∂
∂
+
∂
∂
= −
+
−
Z
y
x f '
y
x x
x
Z
x
y
Z
y
yx f'
y
x
nx
n
n
1
1 nn n
n
f
y
x
yx f '
y
x
nx f
y
x
nZ
+
=
=
−1
∴
∂
∂
+
∂
∂
=x
Z
x
y
Z
y
nZ.
Note 4.4 Euler’s theorem can be extended to a homogeneous function of any number of
variables. Thus, if f(x
1
, x
2
, …, x
n
) be a homogeneous function of degree m in n variables x
1
, x
2
, …, x
n
,
then x
f
x
x
f
x
x
f
x
mf
n
n
n
1
1
1
2
2
2
∂
∂
+
∂
∂
+ +
∂
∂
= .
Example 4.46 If
Z
x y
x y
=
+
−
−
tan
1
3 3
, then prove that x
Z
x
y
Z
y
Z