1. STROUD Worked examples and exercises are in the text Programme F12: Differentiation PROGRAMME F12 DIFFERENTIATION

2. STROUD Worked examples and exercises are in the text Programme F12: Differentiation The gradient of a straight-line graph The gradient of a curve at a given point Algebraic determination of the gradient of a curve Derivatives of powers of x Differentiation of polynomials Derivatives – an alternative notation Standard derivatives and rules Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Newton-Raphson iterative method

3. STROUD Worked examples and exercises are in the text Programme F12: Differentiation The gradient of a straight-line graph The gradient of a curve at a given point Algebraic determination of the gradient of a curve Derivatives of powers of x Differentiation of polynomials Derivatives – an alternative notation Standard derivatives and rules Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Newton-Raphson iterative method

4. STROUD Worked examples and exercises are in the text Programme F12: Differentiation The gradient of a straight-line graph The gradient of the sloping straight line in the figure is defined as: the vertical distance the line rises and falls between the two points P and Q the horizontal distance between P and Q

5. STROUD Worked examples and exercises are in the text Programme F12: Differentiation The gradient of a straight-line graph The gradient of the sloping straight line in the figure is given as:

6. STROUD Worked examples and exercises are in the text Programme F12: Differentiation The gradient of a straight-line graph The gradient of a curve at a given point Algebraic determination of the gradient of a curve Derivatives of powers of x Differentiation of polynomials Derivatives – an alternative notation Standard derivatives and rules Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Newton-Raphson iterative method

7. STROUD Worked examples and exercises are in the text Programme F12: Differentiation The gradient of a straight-line graph The gradient of a curve at a given point Algebraic determination of the gradient of a curve Derivatives of powers of x Differentiation of polynomials Derivatives – an alternative notation Standard derivatives and rules Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Newton-Raphson iterative method

8. STROUD Worked examples and exercises are in the text Programme F12: Differentiation The gradient of a curve at a given point If we take two points P and Q on a curve and calculate, as we did for the straight line, the ratio of the vertical distance the curve rises or falls and the horizontal distance between P and Q the result will depend on the points chosen:

9. STROUD Worked examples and exercises are in the text Programme F12: Differentiation The gradient of a curve at a given point The result depends on the points chosen because the gradient of the curve varies along its length. Because of this the gradient of a curve is not defined between two points as in the case of a straight line but at a single point.

10. STROUD Worked examples and exercises are in the text Programme F12: Differentiation The gradient of a curve at a point P is defined to be the gradient of the tangent at that point: The gradient of a curve at a given point

11. STROUD Worked examples and exercises are in the text Programme F12: Differentiation The gradient of a straight-line graph The gradient of a curve at a given point Algebraic determination of the gradient of a curve Derivatives of powers of x Differentiation of polynomials Derivatives – an alternative notation Standard derivatives and rules Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Newton-Raphson iterative method

12. STROUD Worked examples and exercises are in the text Programme F12: Differentiation The gradient of a straight-line graph The gradient of a curve at a given point Algebraic determination of the gradient of a curve Derivatives of powers of x Differentiation of polynomials Derivatives – an alternative notation Standard derivatives and rules Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Newton-Raphson iterative method

13. STROUD Worked examples and exercises are in the text Programme F12: Differentiation Algebraic determination of the gradient of a curve The gradient of the chord PQ is and the gradient of the tangent at P is

14. STROUD Worked examples and exercises are in the text Programme F12: Differentiation Algebraic determination of the gradient of a curve As Q moves to P so the chord rotates. When Q reaches P the chord is coincident with the tangent. For example, consider the graph of

15. STROUD Worked examples and exercises are in the text Programme F12: Differentiation Algebraic determination of the gradient of a curve At Q: So As Therefore called the derivative of y with respect to x.

16. STROUD Worked examples and exercises are in the text Programme F12: Differentiation The gradient of a straight-line graph The gradient of a curve at a given point Algebraic determination of the gradient of a curve Derivatives of powers of x Differentiation of polynomials Derivatives – an alternative notation Standard derivatives and rules Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Newton-Raphson iterative method

17. STROUD Worked examples and exercises are in the text Programme F12: Differentiation The gradient of a straight-line graph The gradient of a curve at a given point Algebraic determination of the gradient of a curve Derivatives of powers of x Differentiation of polynomials Derivatives – an alternative notation Standard derivatives and rules Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Newton-Raphson iterative method

18. STROUD Worked examples and exercises are in the text Programme F12: Differentiation Derivatives of powers of x Two straight lines Two curves

19. STROUD Worked examples and exercises are in the text Programme F12: Differentiation Derivatives of powers of x Two straight lines (a)

20. STROUD Worked examples and exercises are in the text Programme F12: Differentiation Derivatives of powers of x Two straight lines (b)

21. STROUD Worked examples and exercises are in the text Programme F12: Differentiation Derivatives of powers of x Two curves (a) so

22. STROUD Worked examples and exercises are in the text Programme F12: Differentiation Derivatives of powers of x Two curves (b) so

23. STROUD Worked examples and exercises are in the text Programme F12: Differentiation Derivatives of powers of x A clear pattern is emerging:

24. STROUD Worked examples and exercises are in the text Programme F12: Differentiation The gradient of a straight-line graph The gradient of a curve at a given point Algebraic determination of the gradient of a curve Derivatives of powers of x Differentiation of polynomials Derivatives – an alternative notation Standard derivatives and rules Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Newton-Raphson iterative method

25. STROUD Worked examples and exercises are in the text Programme F12: Differentiation The gradient of a straight-line graph The gradient of a curve at a given point Algebraic determination of the gradient of a curve Derivatives of powers of x Differentiation of polynomials Derivatives – an alternative notation Standard derivatives and rules Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Newton-Raphson iterative method

26. STROUD Worked examples and exercises are in the text Programme F12: Differentiation Differentiation of polynomials To differentiate a polynomial, we differentiate each term in turn:

27. STROUD Worked examples and exercises are in the text Programme F12: Differentiation The gradient of a straight-line graph The gradient of a curve at a given point Algebraic determination of the gradient of a curve Derivatives of powers of x Differentiation of polynomials Derivatives – an alternative notation Standard derivatives and rules Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Newton-Raphson iterative method

28. STROUD Worked examples and exercises are in the text Programme F12: Differentiation The gradient of a straight-line graph The gradient of a curve at a given point Algebraic determination of the gradient of a curve Derivatives of powers of x Differentiation of polynomials Derivatives – an alternative notation Standard derivatives and rules Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Newton-Raphson iterative method

29. STROUD Worked examples and exercises are in the text Programme F12: Differentiation Derivatives – an alternative notation The double statement: can be written as:

30. STROUD Worked examples and exercises are in the text Programme F12: Differentiation The derivative of the derivative of y is called the second derivative of y and is written as: So, if: then Derivatives – an alternative notation

31. STROUD Worked examples and exercises are in the text Programme F12: Differentiation The gradient of a straight-line graph The gradient of a curve at a given point Algebraic determination of the gradient of a curve Derivatives of powers of x Differentiation of polynomials Derivatives – an alternative notation Standard derivatives and rules Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Newton-Raphson iterative method

32. STROUD Worked examples and exercises are in the text Programme F12: Differentiation The gradient of a straight-line graph The gradient of a curve at a given point Algebraic determination of the gradient of a curve Derivatives of powers of x Differentiation of polynomials Derivatives – an alternative notation Standard derivatives and rules Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Newton-Raphson iterative method

33. STROUD Worked examples and exercises are in the text Programme F12: Differentiation Standard derivatives and rules Limiting value of Standard derivatives

34. STROUD Worked examples and exercises are in the text Programme F12: Differentiation Standard derivatives and rules Limiting value of Area of triangle POA is: Area of sector POA is: Area of triangle POT is: Therefore: That is:

35. STROUD Worked examples and exercises are in the text Programme F12: Differentiation Standard derivatives and rules Standard derivatives The table of standard derivatives can be extended to include trigonometric and the exponential functions:

36. STROUD Worked examples and exercises are in the text Programme F12: Differentiation The gradient of a straight-line graph The gradient of a curve at a given point Algebraic determination of the gradient of a curve Derivatives of powers of x Differentiation of polynomials Derivatives – an alternative notation Standard derivatives and rules Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Newton-Raphson iterative method

37. STROUD Worked examples and exercises are in the text Programme F12: Differentiation The gradient of a straight-line graph The gradient of a curve at a given point Algebraic determination of the gradient of a curve Derivatives of powers of x Differentiation of polynomials Derivatives – an alternative notation Standard derivatives and rules Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Newton-Raphson iterative method

38. STROUD Worked examples and exercises are in the text Programme F12: Differentiation Differentiation of products of functions Given the product of functions of x: then: This is called the product rule.

39. STROUD Worked examples and exercises are in the text Programme F12: Differentiation The gradient of a straight-line graph The gradient of a curve at a given point Algebraic determination of the gradient of a curve Derivatives of powers of x Differentiation of polynomials Derivatives – an alternative notation Standard derivatives and rules Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Newton-Raphson iterative method

40. STROUD Worked examples and exercises are in the text Programme F12: Differentiation The gradient of a straight-line graph The gradient of a curve at a given point Algebraic determination of the gradient of a curve Derivatives of powers of x Differentiation of polynomials Derivatives – an alternative notation Standard derivatives and rules Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Newton-Raphson iterative method

41. STROUD Worked examples and exercises are in the text Programme F12: Differentiation Differentiation of a quotient of two functions Given the quotient of functions of x: then: This is called the quotient rule.

42. STROUD Worked examples and exercises are in the text Programme F12: Differentiation The gradient of a straight-line graph The gradient of a curve at a given point Algebraic determination of the gradient of a curve Derivatives of powers of x Differentiation of polynomials Derivatives – an alternative notation Standard derivatives and rules Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Newton-Raphson iterative method

43. STROUD Worked examples and exercises are in the text Programme F12: Differentiation The gradient of a straight-line graph The gradient of a curve at a given point Algebraic determination of the gradient of a curve Derivatives of powers of x Differentiation of polynomials Derivatives – an alternative notation Standard derivatives and rules Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Newton-Raphson iterative method

44. STROUD Worked examples and exercises are in the text Programme F12: Differentiation Functions of a function Differentiation of a function of a function To differentiate a function of a function we employ the chain rule. If y is a function of u which is itself a function of x so that: Then: This is called the chain rule.

45. STROUD Worked examples and exercises are in the text Programme F12: Differentiation Functions of a function Differentiation of a function of a function Many functions of a function can be differentiated at sight by a slight modification to the list of standard derivatives:

46. STROUD Worked examples and exercises are in the text Programme F12: Differentiation The gradient of a straight-line graph The gradient of a curve at a given point Algebraic determination of the gradient of a curve Derivatives of powers of x Differentiation of polynomials Derivatives – an alternative notation Standard derivatives and rules Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Newton-Raphson iterative method

47. STROUD Worked examples and exercises are in the text Programme F12: Differentiation The gradient of a straight-line graph The gradient of a curve at a given point Algebraic determination of the gradient of a curve Derivatives of powers of x Differentiation of polynomials Derivatives – an alternative notation Standard derivatives and rules Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Newton-Raphson iterative method

48. STROUD Worked examples and exercises are in the text Programme F12: Differentiation Newton-Raphson iterative method Tabular display of results Given that x 0 is an approximate solution to the equation f(x) = 0 then a better solution is given as x 1, where: This gives rise to a series of improving solutions by iteration using: A tabular display of improving solutions can be produced in a spreadsheet.

49. STROUD Worked examples and exercises are in the text Programme F12: Differentiation Learning outcomes Determine the gradient of a straight-line graph Evaluate from first principles the gradient of a point on a quadratic curve Differentiate powers of x and polynomials Evaluate second derivatives and use tables of standard derivatives Differentiate products and quotients of expressions Differentiate using the chain rule for a function of a function Use the Newton-Raphson method to obtain a numerical solution to an equation