PARTIAL DIFFERENTIATION

1. Find expressions for all first and second order partial derivatives of the following functions.
   1. z=x^2 + 2x + y
   2. z= y/x^2 + x/y
2. Use the small increments formula to estimate the change in

z=x^2y^4 - x^6 + 4y

when

1. x increases from 1 to 1.1, and y remains fixed at 0
2. x remains fixed at 1, and y decreases from 0 to -0.5

3. Use implicit differentiation to find dy/dx.

x^2 - 2y^3 = 3

What is the value of the derivative at x=1, y= -1?

4. A firms profit function for the production of two goods is given by

pie = 26Q(1) - Q(1)^2 - Q(1)Q(2) - Q(2)^2 +33Q(2) - 43

Find the output levels need to maximize profit. Use second-order derivatives to confirm that the stationary point is a maximum.