# 7 Multivariate Calculus

## 7.1 Partial derivatives

### Practice Problems

Problem 7.1.1: Find the partial derivative with respect to each variable in the problem below:

Problem 7.1.2: Find the partial derivative with respect to each variable in the problem below:

Problem 7.1.3: Find the partial derivative with respect to each variable in the problem below:

Problem 7.1.4: Find the partial derivative with respect to each variable in the problem below:

Problem 7.1.5: Find the partial derivative with respect to each variable in the problem below:

Problem 7.1.6: Find the partial derivative with respect to each variable in the problem below:

Problem 7.1.7: Find the partial derivative with respect to each variable in the problem below:

Problem 7.1.8: The production function for an economy is given below. Find the equation for the marginal productivity of labor and the marginal productivity of capital (MPL and MPK).

Problem 7.1.9: An individual has the choice to consume different quantities of three goods. The utility function is given below. Find the marginal utility equations for each good.

### External Resources

Khan Academy: Graphical Understanding of Partial Derivatives

## 7.2 Comparative statics

### Practice Problems

No practice problems for this section. (Just be able to understand the derivation in the case study from class.)

### External Resources

No external resources for this section.