1. STROUD Worked examples and exercises are in the text PROGRAMME F11 DIFFERENTIATION = slope finding

2. STROUD Worked examples and exercises are in the text 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 Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Second derivatives Newton-Raphson iterative method [optional] Programme F11: Differentiation

3. STROUD Worked examples and exercises are in the text The gradient of a straight-line graph Programme F11: Differentiation The gradient of the sloping line 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

4. STROUD Worked examples and exercises are in the text The gradient (slope) of a straight-line graph Programme F11: Differentiation The gradient of the sloping straight line in the figure is given as:

5. STROUD Worked examples and exercises are in the text 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 Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Second derivatives Newton-Raphson iterative method Programme F11: Differentiation

6. STROUD Worked examples and exercises are in the text The AVERAGE gradient of a curve IN A REGION AROUND a given point P Programme F11: Differentiation What you could call the average gradient of a curve between two points P and Q will depend on the points chosen:

7. STROUD Worked examples and exercises are in the text The gradient of a curve AT a given point The gradient of a curve at a point P is defined to be the gradient of the tangent at that point [= the straight line that intersects the curve only at P, when the curve is not itself a straight line around P - JAB ]: QUESTION: Does a graph always have well-defined tangent at a given point?? Consider e.g. some graphs you’ve drawn in exercises involving the floor function, etc. [JAB] NOTE: If the curve is a straight line around P, the tangent is just that line.

8. STROUD Worked examples and exercises are in the text 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 Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function (Second derivatives –MOVED to a later set of slides) Newton-Raphson iterative method Programme F11: Differentiation

9. STROUD Worked examples and exercises are in the text Algebraic determination of the gradient of a curve Programme F11: Differentiation The gradient of the chord PQ is and the gradient of the tangent at P is

10. STROUD Worked examples and exercises are in the text Algebraic determination of the gradient of a curve Programme F11: Differentiation As Q moves to P so the chord rotates. When Q reaches P the chord is coincident with the tangent.

11. STROUD Worked examples and exercises are in the text 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 Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Newton-Raphson iterative method Programme F11: Differentiation

12. STROUD Worked examples and exercises are in the text Derivatives of powers of x Two straight lines Two curves Programme F11: Differentiation

13. STROUD Worked examples and exercises are in the text Derivatives of powers of x Two straight lines Programme F11: Differentiation (a)

14. STROUD Worked examples and exercises are in the text Derivatives of powers of x Two straight lines (b) Programme F11: Differentiation QUESTION: what about a vertical line, x = d ?? [JAB]

15. STROUD Worked examples and exercises are in the text General definition of the derivative Programme F11: Differentiation dy/dx = limit of  y/  x as  x  0 (from either side)

16. STROUD Worked examples and exercises are in the text Derivatives of powers of x Two curves Programme F11: Differentiation (a) so

17. STROUD Worked examples and exercises are in the text Derivatives of powers of x Two curves Programme F11: Differentiation (b) so

18. STROUD Worked examples and exercises are in the text Derivatives of powers of x A clear pattern is emerging: EXERCISE: Prove this general result, using a result about (a+b) n that we saw when studying combinations. [JAB]

19. STROUD Worked examples and exercises are in the text Algebraic determination of the gradient of a curve At Q: So As Therefore called the derivative of y with respect to x. y = 2x 2 + 5

20. STROUD Worked examples and exercises are in the text 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 Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Newton-Raphson iterative method Programme F11: Differentiation

21. STROUD Worked examples and exercises are in the text Differentiation of polynomials Programme F11: Differentiation To differentiate a polynomial, we differentiate each term in turn:

22. STROUD Worked examples and exercises are in the text 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 Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Newton-Raphson iterative method Programme F11: Differentiation

23. STROUD Worked examples and exercises are in the text Derivatives – an alternative notation Programme F11: Differentiation The double statement: can be written as:

24. STROUD Worked examples and exercises are in the text Towards derivatives of trigonometric functions (JAB) Limiting value of is 1 [NB:  expressed in RADIANS] [in lecture: a rough argument for this] Following slide includes most of a rigorous argument. Programme F11: Differentiation

25. STROUD Worked examples and exercises are in the text Programme F11: Differentiation Area of triangle POA is: Area of sector POA is: Area of triangle POT is: Therefore: That is ((using fact that the cosine tends to 1 -- JAB)):

26. STROUD Worked examples and exercises are in the text Derivatives of trigonometric functions and … Programme F11: Differentiation The table of standard derivatives can be extended to include trigonometric and the exponential functions: [JAB:] The trig cases use the identities for finding sine and cosine of the sum of two angles, and an approximation for the cosine of a small angle (in RADIANS): cos x is approximately 1 – x 2 /2

27. STROUD Worked examples and exercises are in the text 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 Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Newton-Raphson iterative method Programme F11: Differentiation

28. STROUD Worked examples and exercises are in the text Differentiation of products of functions Programme F11: Differentiation Given the product of functions of x: then: This is called the product rule.

29. STROUD Worked examples and exercises are in the text 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 Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Newton-Raphson iterative method Programme F11: Differentiation

30. STROUD Worked examples and exercises are in the text Differentiation of a quotient of two functions Programme F11: Differentiation Given the quotient of functions of x: then: This is called the quotient rule. [BUT I find it easier to use the PRODUCT rule, replacing v by 1/v and using the chain rule below. -- JAB]

31. STROUD Worked examples and exercises are in the text 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 Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function [i.e. compositions of functions] Newton-Raphson iterative method Programme F11: Differentiation

32. STROUD Worked examples and exercises are in the text Functions of a function (compositions) To differentiate a composition w o u we employ the chain rule. If y is a function of u which is itself a function of x so that: y = w(u(x)) e.g. y = sin (x 2 + 1) or y = cos 2 x First, think of this as y = w(u), e.g. y = sin u, with u = x 2 + 1 Then: This is called the chain rule. Programme F11: Differentiation

33. STROUD Worked examples and exercises are in the text Compositions Programme F11: Differentiation Many functions of a function can be differentiated at sight by a slight modification to the list of standard derivatives (F is the u of previous slide):

34. STROUD Worked examples and exercises are in the text For any (differentiable) functions f(x) and g(x), d/dx (f(x) + g(x)) = df(x)/dx + dg(x)/dx d/dx (f(x) - g(x)) = df(x)/dx — dg(x)/dx [and similarly for additions and subtractions of any number of functions] d/dx kf(x) = k df(x)/dx where k is any constant. d/dx x p = p x p-1 where p is any non-zero constant (not just when it is a pos. integer) d/dx (u/v) = d/dx (u.v -1 ) and you can deal with this by the product rule and the power rule just above, instead of remembering the quotient rule separately. Some Clarifications [JAB]

35. STROUD Worked examples and exercises are in the text 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 Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Newton-Raphson iterative method [optional] Programme F11: Differentiation

36. STROUD Worked examples and exercises are in the text Newton-Raphson iterative method [OPTIONAL] Tabular display of results Programme F11: Differentiation 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.