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)