Econ 507 – PS 1
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+5ln(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(x
2y3) 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)
1