Modelling and control by Anthony Rossiter

INTRODUCTION TO DIFFERENTIATION

 

This is a section in the chapter on mathematical skills. Some is revision of pre-university mathematics and some is syllabus mostly covered in year 1 of engineering programmes. Use the left hand toolbar to access the other chapters and themes.

This page gives an introduction to differentiation beginning from a brief discussion of concepts and derivations (which some users can skip) and then from resource 6 onwards a number of worked examples illustrating various common techniques (product and quotient rules, implicit differentation and parametric differentiation).

 

This section contains the following topics. Under each topic there are hardcopy (pdf) notes and a video talk through of key derivations with example problems.

  1. Differentiation 1 - concepts and definitions.
  2. Differentiation 2 - first principles derivations.
  3. Differentiation 3 - superposition
  4. Differentiation 4 - origins of the product rule.
  5. Differentiation 5 - origins of the quotient rule.
  6. Differentiation 6 - worked examples using a lookup table.
  7. Differentiation 7 - worked examples with the product rule
  8. Differentiation 8 - worked examples with the quotient rule.
  9. Differentiation 9 - worked examples with the product and quotient rules.
  10. Differentiation 10 - origins of the chain rule.
  11. Differentiation 11 - worked examples with the chain rule
  12. Differentiation 12 - implicit differentiation.
  13. Differentiation 13 - parametric differentiation.

Differentiation 1 - concepts and definitions.

Introduces the concept of differentiation and the derivative. What do these words mean and what notation is used to represent them. Assumes that viewers are familiar with the terminology of gradient, as for example used with straight line graphs.

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Differentiation 2 - first principles derivations.

Derives the derivatives of some common functions from first principles. This is important background for students as a good understanding of the origins of derivatives will ensure both correct usage and flexibility when something out of the ordinary turns up. Skip straight to resource 6 if you are not interested in seeing an explanation for the properties that fall out of the formal definitions.

Quick test question

Which is true?
A. A derivative is a function defining the gradient of a curve.
B. A derivative is the gradient of a curve at a given point.
C. A derivative is unique for all points on a curve.
D. Several but not all of the above.

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Differentiation 3 - superposition

Superposition is used here to denote a scenario where a function is a sum of several simple functions. The key question is how would we differentiate such a function. The word superposition is used here to denote that one can treat the functions separately and simply add the result. Skip straight to resource 6 if you are not interested in seeing explanation for the properties that fall out of the formal definitions.

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Differentiation 4 - origins of the product rule.

The product rule is to help deal with scenarios where a function is made up as a product of other functions and thus the derivative is not available in a basic table. This resource is focussed on deriving the product rule rather than using it. Usage and examples come later in the series. Skip straight to resource 7 if you are interested in seeing worked examples.

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Differentiation 5 - origins of the quotient rule.

The quotient rule is to help deal with scenarios where a function is made up by dividing different functions and thus the derivative is not available in a basic table. This resource is focussed on deriving the quotient rule rather than using it. Usage and examples come later in the series. Skip straight to resource 8 if you want to see worked examples.

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Differentiation 6 - worked examples using a lookup table.

Assumes that derivatives of common functions are known and stored in a table. Uses several numerical examples to show how the table can be used to find derivatives of commonly encountered functions.

Quick test question

Which are true?
A. The derivative of cos(2x) is 2sin(2x).
B. The derivative of x3+3x is 3(x2+1).
C. The derivative of exp(-3x) is -exp(-3x)/3.
D. None of the above.

Differentiation 7 - worked examples with the product rule.

Assumes that derivatives of common functions are known and stored in a table. Uses several numerical examples to show how the table in combination with the product rule can be used to find derivatives of commonly encountered functions. [OBVIOUS TYPO AROUND 2 min 30-3min 30 where author writes 6x5 rather than 6x2]

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Differentiation 8 - worked examples with the quotient rule.

Assumes that derivatives of common functions are known and stored in a table. Uses several numerical examples to show how the table can be used in combination with the quotient rule to find derivatives of commonly encountered functions. Gives special attention to trigonometric functions, sec, cosec and cot.

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Differentiation 9 - worked examples with the product and quotient rules.

Assumes that derivatives of common functions are known and stored in a table. Uses several numerical examples to show how the table can be used in combination with both the product and the quotient rules to find derivatives of commonly encountered functions.

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Differentiation 10 - origins of the chain rule.

Introduces the chain rule, that is how to differentiate a function of a function. This resource gives a concise derivation or background to the rule so that students understand its origins. Students can skip this if you they just want to use the result.

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Differentiation 11 - worked examples with the chain rule.

This resource gives a number of worked examples using the chain rule to differentiate moderately complicated functions.

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Differentiation 12 - implicit differentiation.

Implicit differentiation is used where there is not a simple explicit expression for the dependent variable, but rather a more complicated expression capturing the relationship between two variables. This resource shows how the chain rule can be used to find the derivative in a straightforward manner. Several examples are included, and notably the rule for differentiating functions containing logarithms and techniques for inverse trigonometric functions. [Obvious typo around 11min where author writes 2x+3 rather than df/dx]

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Differentiation 13 - parametric differentiation.

Shows how to differentiate curves that are described using parametric equations. Gives a very brief derivation of the origins of the approach based on first principles followed by several worked examples.

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