Differentiation

1. Differentiation

2. Differentiation Rules 1.

3. Differentiation Rules 1.

4. Differentiation Rules 2.

5. Differentiation Rules 2.

6. Differentiation Rules 2.

7. Differentiation Rules 3.

8. Differentiation Rules 3.

9. Differentiation Rules for any positive integer n 4.

10. Differentiation Rules for any positive integer n Binominal Theorem 4.

11. Differentiation Rules product rule 5.

12. Differentiation Rules product rule 5.

14. Differentiation Rules where v ≠ 0 quotient rule 6.

15. Differentiation Rules where v ≠ 0 quotient rule 6.

17. Differentiation Rules for any integer n 7.

20. Exercise 13.1 P.34

21. Differentiation of Composite Functions If y is a function of u, and u is a function of x, then y is said to be a composite function of x. If y = u7 and u = 5x2 + 2x + 3, then y is a composite function of x. In fact, in this case, we can write y = (5x2 + 2x + 3)7. (A) Composite Functions

22. Differentiation of Composite Functions If y is a differentiation function of u, and u is a differentiation function of x, then y is said to be a differentiation function of x and (B) Differentiation chain rule

24. Differentiation Rules for any rational number n 8.

27. Exercise 13.2 P.40

28. Differentiation of Inverse Functions and Parametric Functions

29. Differentiation of Inverse Functions and Parametric Functions Given that y = f(x) is a function of x, if it is possible to express x as a function of y in the form x = g(y), then x = g(x) and y = f(x) are inverse functions of each other. (A) Differentiation of Inverse Functions

30. where e.g.

31. Differentiation of Inverse Functions and Parametric Functions Parametric equations are in the form x = f(t) and y = g(t). These equations can be used to describe parametric functions. (B) Differentiation of Parametric Functions

32. where e.g.

33. Exercise 13.3 P.44

34. Differentiation of Implicit Functions When a function between two variables x and y are represented by an equation such that x and y are neither the subject of the equation, then the function is said to be an implicit function of x and y. e.g.

35. Differentiation of Implicit Functions

36. Exercise 13.4 P.46

37. Differentiation of Trigonometric Functions We have learnt that where θ is measured in radians. (A) Differentiation of sine function

39. Differentiation of Trigonometric Functions (B) Differentiation of cosine function

40. Differentiation of Trigonometric Functions (C) Differentiation of tangent function

41. Differentiation of Trigonometric Functions (D) Differentiation of cotangent function

42. Differentiation of Trigonometric Functions (E) Differentiation of sec function

43. Differentiation of Trigonometric Functions (F) Differentiation of cosec function

44. Exercise 14.1 P.59

45. Differentiation of Trigonometric Functions with Composite Function or Implicit Function or Parametric Function

46. Find of the following.

47. Second Derivatives If y = f ’(x) is a differentiable function of x, the derivative of y = f ’(x) with respect to x is called the second derivative of y = f(x) with respect to x and is denoted by or . That is

48. Exercise 14.2 P.64