# 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:

Answer:

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

Answer:

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

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

Answer:

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

Answer:

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

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

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**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 (MP_{L} and MP_{K}).

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**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.

Answers:

### External Resources

Khan Academy: Partial Derivatives, Introduction

Khan Academy: Graphical Understanding of Partial Derivatives

Khan Academy: Gradients

## 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.