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# 3.1. Multivariate Calculus: Partial derivatives

### Learning Objectives

You should be able to

• understand what a multivariate function is
• evaluate a multivariate function at a point
• draw level curves of a function of two variables
• compute first order partial derivatives
• compute higher order direct and mixed partial derivatives
• understand how many partial derivatives of each order a multivariate function has

### Get Started: What to do next

If you can pass the diagnostic test, then we believe you are ready to move on from this topic.
Our recommendation is to:
1. Test your ability by trying the diagnostic quiz. You can reattempt it as many times as you want and can leave it part-way through.
2. Identify which topics need more work. Make a note of any areas that you are unable to complete in the diagnostic quiz, or areas that you don't feel comfortable with.
3. Watch the tutorials for these topics, you can find these below. Then try some practice questions from the mini quiz for that specific topic. If you have questions, please ask on the forum.
4. Re-try the diagnostic quiz. And repeat the above steps as necessary until you can pass the diagnostic test. A pass grade is 80%.

Optionally, also see if you can apply these skills to an economic application or look at the additional resources and advanced quizzes at the bottom of the page,

### Tutorials: How to guides

#### 3.1.1. Functions of several variables

An introduction to multivariate functions and level curves $$\mathscr{L}_k = \{(x,y) \mid f(x,y)=k\}$$

#### 3.1.2. First order partial derivatives

$$\frac{\partial f}{\partial x} (a,b) =\left. \frac{\partial }{\partial x} (f) \right|_{(x,y)=(a,b)} = f'_x(a,b) =\left.f_x(x,y)\right|_{(a,b)} = f'_1(a,b)$$

#### 3.1.3. Higher order partial derivatives

direct: $$\frac{\partial^2 f}{\partial x^2} =\frac{\partial }{\partial x} \left(\frac{\partial }{\partial x} (f) \right)= f''_{xx}$$,      mixed: $$\frac{\partial^2 f}{\partial y\partial x} =\frac{\partial }{\partial y} \left(\frac{\partial }{\partial x} (f) \right)= f''_{xy}$$

### Economic Application: Production functions, isoquants, and marginal products

In economics, production technology is usually modelled by the means of a production function - a (usually multivariate) function describing how quantities of inputs used are mapped into output. This application introduces production functions as a modelling device, as well as the important concept of isoquants (contours of the production function) and marginal products (partial derivatives of the production function).

#### Economic Application Exercise

The following quiz allows you to test your understanding of production functions, isoquants and marginal products. In addition, the second question introduces two important classes of contour lines in the context of consumer theory - indifference curves and budget lines.

### Further practice & resources

You can find practice quizzes on each topic in the links above the tutorial videos, or you can take the diagnostic quiz as many times as you like to. If you want to test yourself further, then try the advanced quiz linked below.

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