MATHEMATICAL ECONOMICS

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 + (2x3 − 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)

1