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3.2. Multivariate Calculus: Implicit differentiation

 3.2.0. Introduction: An economist's perspective

 Learning Objectives

You should be able to

  • use implicit differentiation to find the derivative of one variable which is implicitly defined in terms of another via an equation
  • use partial derivatives to compute derivatives of a function defined in terms of another via a level curve.

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

3.2.1. Implicit differentiation

Differentiate both sides of an equation and rearrange to compute the derivative of a variable which is implicitly defined in terms of another. 
If \(y\) is defined in terms of \(x\) via the level curve \(f(x,y)=k\), then  \( \frac{\mathrm{d}y}{\mathrm{d}x} = -\frac{f'_x}{f'_y} \)

Slides 3.2.1  Mini quiz 3.2.1


 Economic Application: Marginal rate of substitution

This application introduces the concept of utility functions, indifference curves (the contour plots of utility functions), and marginal rates of substitution. In particular, it is shown that using implicit differentiation, the marginal rate of substitution exhibited by a utility function can be expressed as a ratio of marginal utilities - something that you will see often in your studies.

 3.2.2. Economic Application Example

Slides 3.2.2

 Economic Application Exercise 

The following quiz allows you to test your understanding of the mathematics and economics behind the concept of marginal rate of substitution. Further, the quiz includes a common example of a utility maximization problem which can be solved by using implicit differentiation.

Economic Applications Quiz 3.2 

  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.

Advanced quiz 3.2  Further links and resources 3

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