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2.3. Univariate Calculus: Rules of Differentiation

 2.3.0. Introduction: An economist's perspective

 Learning Objectives

You should be able to differentiate the following types for function

  • power functions like  \(x^4\)
  • exponential functions like \(e^{3x+2}\)
  • logarithm functions like \(\ln(5x+7)\)
You should also be able to apply the following rules of differentiation, possibly using several in combination to differentiate functions made from those above.
  • the sum rule (for addition)
  • the product rule (for multiplication)
  • the quotient rule (for division)
  • the chain rule (for functions of functions)

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


2.3.1. The power rule

\(\frac{\mathrm{d}}{\mathrm{d}x} x^n = nx^{n-1}\)

Slides 2.3.1  Mini quiz 2.3.1


2.3.2. Natural exponential and logarithm

\(\frac{\mathrm{d}}{\mathrm{d}x} e^{ax} = ae^{ax}, \frac{\mathrm{d}}{\mathrm{d}x} \ln{(ax+b)} = \frac{a}{ax+b}\)

Slides 2.3.2 Mini quiz 2.3.2


2.3.3. The sum rule

\((f+g)' = f'+g'\)

Slides 2.3.3 Mini quiz 2.3.3


2.3.4. The product rule

\((fg)' = f'g+fg'\)

Slides 2.3.4 Mini quiz 2.3.4


2.3.5. The quotient rule

\(\left(\frac{u}{v}\right) = \frac{u'v-v'u}{v^2}\) 

Slides 2.3.5 Mini quiz 2.3.5


2.3.6. The chain rule

\(\frac{\mathrm{d}}{\mathrm{d}x} f(g(x)) = f'(g(x)) g'(x)\)

Slides 2.3.6 Mini quiz 2.3.6


2.3.7. Combining differentiation rules

Differentiate more complicated functions by combining the rules above.

Slides 2.3.7 Mini quiz 2.3.7


 Economic Application: Marginal cost and marginal revenue

Producing output is costly and the relationship between the quantity produced and the cost of production is described by a producer's cost function . On the other hand by producing and selling output the producer realizes revenue and the relationship between the quantity produced and the revenue is described by a producer's revenue function. In deciding how much to produce, a producer should consider if the additional revenue from an extra unit produced exceeds or falls short of the additional cost from producing this extra unit. These rates of change are respectively measured by the marginal cost and the marginal revenue functions. The following application considers how these functions can be obtained by using the rules of differentiation and provides an economic interpretation of the meaning of derivatives.

 2.3.8. Economic Application Example

Slides 2.3.8

 Economic Application Exercise

The following quiz allows you to test your understanding of using the rules for differentiation to obtain marginal cost functions from a total cost function, a marginal revenue function from a demand function, and finding the elasticity of certain functions.

Economic Applications Quiz 2.3

 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 2.3  Further links and resources 2

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