Department of Economics
University of Maryland
Economics 325
Intermediate Macroeconomic Analysis
Practice Problem Set 1 Suggested Solutions
Professor Sanjay Chugh
Spring 2011
1.
Partial Derivatives.
For each of the following multi-variable functions, compute the
partial derivatives with respect to both
x
and
y
.
Solution:
In order to compute the partial derivative with respect to
x,
momentarily
pretend that
y
is a constant (for example, imagine momentarily that
y
= 5) and
proceed to differentiate using the usual rules of calculus.
Likewise, in order to
compute the partial derivative with respect to
y
, momentarily pretend that
x
is a
constant (for example, imagine momentarily that
x
= 5) and proceed to differentiate
using the usual rules of calculus.
Applying this algorithm to each of the given functions:
a.
( ,
)
f x y
xy
We have
( ,
)
x
f
x y
y
and
( ,
)
y
f
x y
x
.
b.
( ,
)
2
3
f x y
x
y
±
We have
( ,
)
2
x
f
x y
and
( ,
)
3
y
f
x y
.
c.
2
4
( ,
)
f x y
x y
We have
4
( ,
)
2
x
f
x y
xy
and
2
3
( ,
)
4
y
f
x y
x y
.
d.
( ,
)
ln
2ln
f x y
x
y
±
We have
( ,
)
1/
x
f
x y
x
and
( ,
)
2/
y
f
x y
y
.
e.
( ,
)
2
2
f x y
x
y
±
Recall from principles of basic mathematics that we can write this function as
1/ 2
1/ 2
( ,
)
2
2
f x y
x
y
±
.
Hence, the partial derivatives are
1/ 2
( ,
)
1/
x
f
x y
x
x
²
and
1/ 2
( ,
)
1/
y
f
x y
y
y
²
.