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Basic Skills in Higher Mathematics

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  1. Basic Skills in Higher Mathematics Mathematics 1(H)Outcome 3 Robert GlenAdviser in Mathematics

  2. Mathematics 1(Higher) Outcome 2 Use basic differentiation Differentiation dydx f (x)

  3. Mathematics 1(Higher) Outcome 2 Use basic differentiation PC Index Click on the PC you want PC(a) Basic differentiation dydx f (x) PC(b) Gradient of a tangent PC(c) Stationary points

  4. Mathematics 1(Higher) Outcome 3 Use basic differentiation PC(a) Differentiate a function reducible to a sum of powers of x PC(a) - Basicdifferentiation dydx f (x)

  5. Mathematics 1(Higher) Outcome 3 Use basic differentiation PC(a) Differentiate a function reducible to a sum of powers of x PC(a) - Basic differentiation Click on the section you want 1Simple functions 2Simple functions multiplied by a constant 3Negative indices 4Fractional indices 5Negative and fractional indices with constant 6Sums of functions (simple cases) 7Sums of functions (negative indices) 8Sums of functions ( algebraic fractions)

  6. Mathematics 1(Higher) Outcome 3 Use basic differentiation PC(a) Differentiate a function reducible to a sum of powers of x PC(a) - Basic differentiation 1Simple functions dydx f (x)

  7. Mathematics 1(Higher) Outcome 3 Use basic differentiation PC(a) Differentiate a function reducible to a sum of powers of x Some examples Every function f(x) has a related function called the derived function.The derived function is written f(x) “f dash x” f(x) f(x) x3 3x2 x6 6x5 The derived function is also called the derivative. x10 10x9 x (= x1) 1 To find the derivative of a functionyou differentiate the function. 0 3 (= 3x0)

  8. Mathematics 1(Higher) Outcome 3 Use basic differentiation PC(a) Differentiate a function reducible to a sum of powers of x Rule No. 1 for differentiation If f(x) = xn , then f (x) = nxn -1

  9. Mathematics 1(Higher) Outcome 3 Use basic differentiation PC(a) Differentiate a function reducible to a sum of powers of x Here are the answers Differentiate each of these functions. 1f(x) = x4 1f (x) = 4x3 2f(x) = x5 2f (x) = 5x4 3g(x) = 8x7 3g(x) = x8 4h(x) = x2 4h(x) = 2x 5f (x) = 12x11 5f(x) = x12 6f (x) = 1 6f(x) = x 7f (x) = 0 7f(x) = 5

  10. Mathematics 1(Higher) Outcome 3 Use basic differentiation PC(a) Differentiate a function reducible to a sum of powers of x End of Section 1 Continue with Section 2

  11. Mathematics 1(Higher) Outcome 3 Use basic differentiation PC(a) Differentiate a function reducible to a sum of powers of x PC(a) - Basic differentiation 2Simple functionsmultiplied by a constant dydx f (x)

  12. Mathematics 1(Higher) Outcome 3 Use basic differentiation PC(a) Differentiate a function reducible to a sum of powers of x More examples f(x) f(x) 6x2 2x3 2  3x2 18x5 3x6 3  6x5 2x10 20x9 2  10x9 5x 5  1 5

  13. Mathematics 1(Higher) Outcome 3 Use basic differentiation PC(a) Differentiate a function reducible to a sum of powers of x Rule No. 2 for differentiation If f(x) = axn , then f (x) = anxn -1

  14. Mathematics 1(Higher) Outcome 3 Use basic differentiation PC(a) Differentiate a function reducible to a sum of powers of x Here are the answers Differentiate each of these functions. 1f(x) = 3x4 1f (x) = 12x3 2f(x) = 2x5 2f (x) = 10x4 3g(x) = 4x7 3g(x) = ½ x8 4h(x) = 5x2 4h(x) = 10x 5f (x) = 3x11 5f(x) = ¼ x12 6f (x) = 8 6f(x) = 8x 6f (x) = 0 7f(x) = 10

  15. Mathematics 1(Higher) Outcome 3 Use basic differentiation PC(a) Differentiate a function reducible to a sum of powers of x End of Section 2 Continue with Section 3

  16. Mathematics 1(Higher) Outcome 3 Use basic differentiation PC(a) Differentiate a function reducible to a sum of powers of x PC(a) - Basic differentiation 3Negative indices dydx f (x)

  17. Mathematics 1(Higher) Outcome 3 Use basic differentiation PC(a) Differentiate a function reducible to a sum of powers of x Rule No. 1 for differentiation If f(x) = xn , then f (x) = nxn -1

  18. Mathematics 1(Higher) Outcome 3 Use basic differentiation PC(a) Differentiate a function reducible to a sum of powers of x More examples 1f(x) = -4 -3 = x -3 -1 x -4 f (x) = -3 x? -5 -4 -3 -2 -1 0 = Note:This is an example of using Rule No.1with a negative index.

  19. Mathematics 1(Higher) Outcome 3 Use basic differentiation PC(a) Differentiate a function reducible to a sum of powers of x More examples 2f(x) = -1 -2 = x -1 -1 f (x) = -1 x? x -2 -2 -1 0 = Note:This is an example of using Rule No.1with a negative index.

  20. Mathematics 1(Higher) Outcome 3 Use basic differentiation PC(a) Differentiate a function reducible to a sum of powers of x Here are the answers Differentiate each of these functions. 1f(x) = x -2 1f (x) = -2x -3 2f(x) = x -4 2f (x) = -4x -5 3g(x) = -5x -6 = 3g(x) = 4h(x) = -1x -2 = 4h(x) = 5f(x) = 5f (x) = -10x -11 =

  21. Mathematics 1(Higher) Outcome 3 Use basic differentiation PC(a) Differentiate a function reducible to a sum of powers of x End of Section 3 Continue with Section 4

  22. Mathematics 1(Higher) Outcome 3 Use basic differentiation PC(a) Differentiate a function reducible to a sum of powers of x PC(a) - Basic differentiation 4Fractional indices dydx f (x)

  23. Mathematics 1(Higher) Outcome 3 Use basic differentiation PC(a) Differentiate a function reducible to a sum of powers of x Rule No. 1 for differentiation If f(x) = xn , then f (x) = nxn -1

  24. Mathematics 1(Higher) Outcome 3 Use basic differentiation PC(a) Differentiate a function reducible to a sum of powers of x More examples 3f(x) = x? f (x) = -1 = -1 0 1 Note:This is an example of using Rule No.1with a fractional index.

  25. Mathematics 1(Higher) Outcome 3 Use basic differentiation PC(a) Differentiate a function reducible to a sum of powers of x More examples 4f(x) = x? f (x) = -1 = -1 0 1 Note:This is an example of using Rule No.1with a fractional index.

  26. Mathematics 1(Higher) Outcome 3 Use basic differentiation PC(a) Differentiate a function reducible to a sum of powers of x Here are the answers Differentiate each of these functions. 1f(x) = 1f (x) = 2f(x) = 2f (x) = 3g(x) = 3g(x) = 4h(x) = 4h(x) =

  27. Mathematics 1(Higher) Outcome 3 Use basic differentiation PC(a) Differentiate a function reducible to a sum of powers of x End of Section 4 Continue with Section 5

  28. Mathematics 1(Higher) Outcome 3 Use basic differentiation PC(a) Differentiate a function reducible to a sum of powers of x PC(a) - Basic differentiation 5Negative and fractional indices with constant dydx f (x)

  29. Mathematics 1(Higher) Outcome 3 Use basic differentiation PC(a) Differentiate a function reducible to a sum of powers of x Rule No. 2 for differentiation If f(x) = axn , then f (x) = anxn -1

  30. Mathematics 1(Higher) Outcome 3 Use basic differentiation PC(a) Differentiate a function reducible to a sum of powers of x More examples 5f(x) = -4 -3 = 2x -3 -1 x -4 f (x) = -6 x? -5 -4 -3 -2 -1 0 = Note:This is an example of using Rule No.2with a negative index.

  31. Mathematics 1(Higher) Outcome 3 Use basic differentiation PC(a) Differentiate a function reducible to a sum of powers of x More examples 6f(x) = -1 -2 = 5x -1 -1 f (x) = -5 x? x -2 -2 -1 0 = Note:This is an example of using Rule No.2with a negative index.

  32. Mathematics 1(Higher) Outcome 3 Use basic differentiation PC(a) Differentiate a function reducible to a sum of powers of x Here are the answers Differentiate each of these functions. 1f(x) = 3x -2 1f (x) = -6x -3 2f(x) = 5x -4 2f (x) = -20x -5 3g(x) = -10x -6 = 3g(x) = 4h(x) = -2x -2 = 4h(x) = 5f(x) = 5f (x) = -30x -11 =

  33. Mathematics 1(Higher) Outcome 3 Use basic differentiation PC(a) Differentiate a function reducible to a sum of powers of x More examples 7f(x) = x? f (x) = 3 6  ½ = 3 -1 = -1 0 1 Note:This is an example of using Rule No.2with a fractional index.

  34. Mathematics 1(Higher) Outcome 3 Use basic differentiation PC(a) Differentiate a function reducible to a sum of powers of x More examples 8f(x) = x? 9  2/3 = 6 f (x) = -1 = -1 0 1 Note:This is an example of using Rule No.2with a fractional index.

  35. Mathematics 1(Higher) Outcome 3 Use basic differentiation PC(a) Differentiate a function reducible to a sum of powers of x Here are the answers Differentiate each of these functions. 1f(x) = 1f (x) = 2f(x) = 2f (x) = 3g(x) = 3g(x) = 4h(x) = 4h(x) =

  36. Mathematics 1(Higher) Outcome 3 Use basic differentiation PC(a) Differentiate a function reducible to a sum of powers of x When a function is given in the form of an equation, the derivative is written in the form = f (x) Examples 2f(x) = 4x -3f (x) = -12x -4 1f(x) = 3x2f (x) = 6x y = 3x2 y = 4x -3 = -12x -4 = 6x

  37. Mathematics 1(Higher) Outcome 3 Use basic differentiation PC(a) Differentiate a function reducible to a sum of powers of x End of Section 5 Continue with Section 6

  38. Mathematics 1(Higher) Outcome 3 Use basic differentiation PC(a) Differentiate a function reducible to a sum of powers of x PC(a) - Basic differentiation 6Sums of functions (simple cases) dydx f (x)

  39. Mathematics 1(Higher) Outcome 3 Use basic differentiation PC(a) Differentiate a function reducible to a sum of powers of x More examples g(x) h(x) k(x) 1 2 f(x) = 5x2 - 3x + 1 g(x) = (x + 3)(x - 2) = x2 + x - 6 10x - 3 + 0 f (x) = g (x) = = 10x - 3 2x + 1

  40. Mathematics 1(Higher) Outcome 3 Use basic differentiation PC(a) Differentiate a function reducible to a sum of powers of x Rule No. 3 for differentiation If f(x) = g(x) + h(x) + k(x) +……..., then f (x) = g (x) + h (x) +k (x) +..

  41. Mathematics 1(Higher) Outcome 3 Use basic differentiation PC(a) Differentiate a function reducible to a sum of powers of x Here are the answers Differentiate each of these functions. 1f(x) = x2 + 7x - 3 1f (x) = 2x + 7 2f(x) = 3x2 - 4x + 10 2f (x) = 6x - 4 3g(x) = 3x2 + 2x - 5 3g(x) = x(x2 + x- 5) 4h(x) = 4x3 - 10x2 4h(x) = 12x2 - 20x 5f (x) = 35x4 - 35x6 5f(x) = x5(7- 5x2) 6f (x) = 3x2 + 2x + 1 6f(x) = x3 + x2 + x + 1 7f (x) = x3 + x 7f(x) = ¼ x4 +½ x2

  42. Mathematics 1(Higher) Outcome 3 Use basic differentiation PC(a) Differentiate a function reducible to a sum of powers of x End of Section 6 Now do Section A2 on page 33 of the Basic Skills booklet

  43. Mathematics 1(Higher) Outcome 3 Use basic differentiation PC(a) Differentiate a function reducible to a sum of powers of x PC(a) - Basic differentiation 7Sums of functions (negative indices) dydx f (x)

  44. Mathematics 1(Higher) Outcome 3 Use basic differentiation PC(a) Differentiate a function reducible to a sum of powers of x More examples 1 2 - 3x - y = f(x) = - 1 + 5x -1 = 2x -2 - 3x - x -1 - = x -2 1 + f (x) = -4x-3 - 3 + 5x-2 + 0 x -2 -2x-3 = + = - 3 - + - = +

  45. Mathematics 1(Higher) Outcome 3 Use basic differentiation PC(a) Differentiate a function reducible to a sum of powers of x Here are the answers Differentiate each of these functions. - 1f(x) = 1f (x) = - 2 - 2x + x2 - x 2y = = - 2 + 2x - 1 - + 3x2 3y = - = 3 + + 6x - - - 4g(x) = + 4g(x) = +

  46. Mathematics 1(Higher) Outcome 3 Use basic differentiation PC(a) Differentiate a function reducible to a sum of powers of x End of Section 7 Continue with Section 8

  47. Mathematics 1(Higher) Outcome 3 Use basic differentiation PC(a) Differentiate a function reducible to a sum of powers of x PC(a) - Basic differentiation 8Sums of functions (algebraic fractions) dydx f (x)

  48. Mathematics 1(Higher) Outcome 3 Use basic differentiation PC(a) Differentiate a function reducible to a sum of powers of x More examples 1 2 y = f(x) = - + = = + - = 1 + 3x -1 - 2x -2 - 3x -1 + 5 = x + 4x -3 0 - 3x -2 = f  (x) = 1 3x -2 + + 0 - = + = 1 +

  49. Mathematics 1(Higher) Outcome 3 Use basic differentiation PC(a) Differentiate a function reducible to a sum of powers of x Here are the answers Differentiate each of these functions. 1f(x) = 1f (x) = 2x - 2 2y = 2 1 = 3g(x) = 1+ 3g(x) = - + = 4y = 4

  50. Mathematics 1(Higher) Outcome 3 Use basic differentiation PC(a) Differentiate a function reducible to a sum of powers of x End of Section PC(a) Now do Section A3 on page 33 of the Basic Skills booklet