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Econ 507 – PS
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ECON 507- Mathematical Economics
Problem Set 1
Partial Derivatives
Differentiate the following functions with respect to x and y (ie. calculate
∂f
∂x
and
∂f
∂
y
):
1. f(x, y) = 6x + 8y − xy
2. f(x, y) = x
2
y
3. f(x, y) = (xy)2 + (2×3 − 7y)(ln y − ex)
4. f(x, y) =
xy+5
ln(x)
5. f(x, y) =
√
e2xy − ln(xy)
6. f(x, y) = ln(e2x+5y
2
+ 10x)
7. f(x, y) =
(
x
1
2 + y
1
2
)2
8. f(x, y) =
ln(x2y3)
xy
9. f(x, y) = yx
Second Order Partial Derivatives
Second order partial derivatives are calculated the same way that partial derivatives are – treat any
other variables other than the one you are differentiating with respect to as constants. If we use fx
to denote
∂f
∂x
and fy to denote
∂f
∂y
, then the second order derivatives are calculated as
∂fx
∂x
(which
we can write as fxx to save space),
∂fx
∂y
(which we can write as fxy),
∂fy
∂y
(which we can write as
fyy), and
∂fy
∂x
(which we can write as fyx). For the following functions, compute all second order
derivatives fxx, fxy, fyy, and fyx.
1. f(x, y) = (x + y)2
2. f(x, y) = x
1
2 y
1
2
3. f(x, y) = ln(x + y)
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