1. Analytical Toolbox Differential calculus By Dr J.P.M. Whitty

2. 2 Learning objectives After the session you will be able to: After the session you will be able to: Define the derivative in terms of a imitating functionDefine the derivative in terms of a imitating function Differentiate simple algebraic functions from first principlesDifferentiate simple algebraic functions from first principles Apply rules of differentiationApply rules of differentiation Use math software to solve simple differentiation problemsUse math software to solve simple differentiation problems

3. 3 Differential Calculus Calculus is the mathematics of change A point in three dimensions needs six pieces of information to be fully described. Calculus is the mathematics of change A point in three dimensions needs six pieces of information to be fully described. We use the Derivative Function, f, to describe the rate of change of the original function f.We use the Derivative Function, f, to describe the rate of change of the original function f.

4. 4 Example Find the derivative (using first principles) of f (x) = x 2 - 8x + 9. Here we use the definition: Where: Giving:

5. 5 Example Cont: Simplifying the denominator: That is: Cancel out the h’s 0

6. 6 Differentiation of standard functions It is usual in calculus textbooks that to see tables of standard functions and their respective differential coefficients. For this introductory course you will only require the following For this introductory course you will only require the following

7. 7 Class Examples Time Copy and complete the following table.

8. 8 Class examples Differentiate the following:

9. 9 Another example Find the first and second derivatives of:

10. 10 Class Examples Time Find the first and second derivative of:

11. 11 Differentiation Formulae Several ‘short-cut’ rules exist for differentiation, eliminating the need for tedious and repetitive use of limit expressions. Proofs of these can be found in most calculus books.

12. 12 Revision Questions 1. Complete the following table:

13. 13 More Revision Qs 1. Differentiate the following: 2. Find the first and second derivatives of:

14. 14 Revision Qs (solutions) 1. Complete the following table:

15. 15 More Revision Qs (Solutions) 1. Differentiate the following: 2. The first and second derivatives

16. 16 The Product rule: This is used when we have two primitive functions multiplied together. If then

17. 17 The Product rule: Proof Here we employ the definition of the derivative thus: Re-writing this gives

18. 18 The Product rule: Proof Now use the fact that the terms in parenthesis are derivatives of the individual functions, to complete the proof, i.e.: Hence Substitution gives: Evaluation of the limit renders the required result

19. 19 The Product rule: Proof (Alternative nomenclature) In many calculus text books, the less formal functions of u and v are used. This makes the preceding proof a little less tedious, as follows If then 0

20. 20 Lemma 1: When using the calculus we rarely resort to using formulae, we tend to remember a mechanism of a pattern, these leads us to the lemma. When using the calculus we rarely resort to using formulae, we tend to remember a mechanism of a pattern, these leads us to the lemma. Lemma 1:The first times the diff. of the second-plus-the second times the diff. of the first

21. 21 Example If find FirstSecond Diff of the second Diff of the first

22. 22 Another Example: If find Ans:

23. 23 if The quotient rule: This is used when we have two primitive functions dividend by one another Then or

24. 24 Lemma 2: Again we tend to remember a mechanism of a pattern, this leads us to the lemma. Lemma 1:The bottom times the diff. of the top-minus-the top times the diff. of the bottom – ALL OVER THE BOTTOM SQUARED In a sense this is simply the reverse of the product rule and then dividing by the square of the bottom!

25. 25 Example If find bottom top Diff of the topDiff of the bottom bottom squared

26. 26 Alternative approach An alternative approach is to use logarithmic differentiation as follows: Take logs Differentiate using the chain rule as appropriate:

27. 27 Alternative approach The logarithmic approach is especially useful when the primitive functions are themselves products (or quotients), the details of the method are left to P7 of the recommended reading. But here we shall consider another quick example, i.e.:

28. 28 Logarithmic differentiation The approach is always the same take natural logs, remembering ever primitive above the line is positive and everything below is negative. Then differentiate the resulting primitive functions, remembering that the RHS is equal to the 1/y multiplied by the differential coefficient. The rest is just algebra!

29. 29 Logarithmic differentiation Example Taking logs of both sides gives: Differentiate: Substitute:

30. 30 Use of mathematic Software As the expressions get more and more complicated the methods stay the same but the algebra gets more and more tedious. Thankfully these days MATLAB has the answer that is the symbolic toolbox we have been using in previous lessons can be used to differentiate complicated functions such as that in the last example.

31. 31 MATLAB: Differentiation The process is the same as usual. i.e. easy as ABC! A.Set up your symbolics in MATLAB using the syms command B.Type in the expression remembering the rules of BIDMAS C.Use the appropriate MATLAB function in this case diff( ), making pretty if required.

32. 32 MATLAB Solution The commands are very straight forward but you may have to do a little bit of algebra at the end to get the result in the same form alternatively use the MATLAB commands to do it for you!!

33. 33 MATLAB Solution cont… For instance try factorizing the resulting expression using the factor function we have used in class previously, thus

34. 34 Summary Have we met our learning objectives? Specifically: are you able to: Define the derivative in terms of a imitating functionDefine the derivative in terms of a imitating function Differentiate simple algebraic functions from first principlesDifferentiate simple algebraic functions from first principles Apply rules of differentiationApply rules of differentiation Use math software to solve simple differentiation problemsUse math software to solve simple differentiation problems

35. 35 Homework 1. Find the derivative of 2. Find the derivative of 3. Find the derivative of

36. 36 Examination type questions 1. Differentiate the following with respect to x. a. b. b. c. c. d. Explain how you would use math software to verify your results

37. 37 Examination type questions 2. Evaluate the tangent to the curve at a point P(2,6) and find the equation of the tangent of that line. Given that the product of the normal and tangential line gradients is minus unity find the equation of the normal.

38. 38 More Calculus WELL DONE!! You have now completed the first part of this learning pack. Have a break and then you can try the next part where you will be introduced to the inverse process to differentiation, namely: learning packlearning pack Integration