1. Higher Unit 3 Further Differentiation Trig Functions Further Integration Integrating Trig Functions Differentiation The Chain Rule

2. The Chain Rule for Differentiating To differentiate composite functions (such as functions with brackets in them) use:

3. The Chain Rule for Differentiating In practice we do this as follows: 1. Differentiate the bracket: 2. Differentiate inside the bracket: 3. Multiply the answers together

4. Differentiate Use indices to get rid of root sign Chain Rule Simplify

5. Differentiate Use indices to get variable on working line Chain Rule Simplify

6. Differentiate Use indices to get rid of root sign Chain Rule Simplify

7. First bring denominator on to the working line (indices) 1. Differentiate the bracket: 2. Differentiate inside the bracket: 3. Multiply the answers together

8. First use indices to get rid of square root and bring denominator on to the working line 1. Differentiate the bracket: 2. Differentiate inside the bracket: 3. Multiply the answers together

9. Differentiate (5x – 1) 3 Chain Rule Simplify 3(5x – 1) 2 × 5 15(5x – 1) 2 Differentiate (2x 3 – x + 2) 4 Chain Rule 4 (2x 3 – x + 2) 3 (6x 2 – 1)

10. Differentiate Use indices to get rid of root sign Chain Rule Simplify

11. Differentiate Use indices to get variable on working line Chain Rule Simplify

12. 1.2.3.

13. In a small factory the cost, C, in pounds of assembling x components in a month is given by: Calculate the minimum cost of production in any month, and the corresponding number of components that are required to be assembled. SP’s  C ’ (x) = 0

14. C ’ (x) = 0  Number of components must be positive Shape→5→x– 0 +    When x = 5 C’(x)C’(x)

15. Derivatives of Trig Functions f(x) = sin x  f ’ (x) = cos x f(x) = cos x  f ’ (x) = – sin x To calculate the value of any derivative the angles must be measured in radians The basic derivatives are given in a formula list in the exam

16. Differentiate Put in brackets use Chain Rule Simplify

17. Find f ’ ( π / 3 ) when Put in brackets use Chain Rule Simplify sin 3 x is the same as (sin x) 3

18. Differentiate Put in brackets use Chain Rule Simplify

19. Differentiate Use Chain Rule Simplify

20. Differentiate Use Chain Rule Simplify

21. Differentiate Use Chain Rule Simplify

22. Integrating Composite Functions (4x – 1) 7 4  (6 + 1) + c

23. Integrating Composite Functions Use indices to get denominator on working line

26. Evaluate

27. Evaluate f(x) given f ’ (x) = (2x – 1) 3 and f(1) = 2 Since f(1) = 2

28. Find p, given

29.  3 Reverse process

30. A curve for which passes through (–1, 2). Express y in terms of x. Curve passes through (–1, 2)

31. Given the acceleration a is: If it starts at rest, find an expression for the velocity v. When t = 0, v = 0

32. Integrating Trig Functions Integration is opposite of differentiation

33. Special Trigonometry Integrals

36. Area between Trig Curves A The diagram shows the graphs of y = –sin x and y = cos x a)Find the coordinates of A b)Hence find the shaded area C A S T 0o0o0o0o 180 o 270 o 90 o Curves intersect where y = y, ie – sin x = cos x Divide through by – cos x tan x = – 1 tan -1 (1) = 45 o ( π / 4 ) x = 3π / 4 or 7π / 4 A( 3π / 4, ?) A( 3π / 4, – 1 / √2 ) 135 o 315 o

37. A b) Hence find the shaded area ( 3π / 4, – 1 / √2 ) Area = ∫ (top curve – bottom curve) ∫ (top curve – bottom curve)

38. By writing cos 3x as cos(2x + x), show that cos 3x = 4cos 3 x – 3cos x Hence find ∫ cos 3 x dx From cos(A + B) = cos A cos B – sin A sin B cos(2x + x) = cos 2x cos x – sin 2x sin x = (2cos 2 x – 1)cos x – (2sin x cos x) sin x = 2cos 3 x – cos x – 2sin 2 x cos x = 2cos 3 x – cos x – 2(1 – cos 2 x) cos x = 2cos 3 x – cos x – 2cos x + 2cos 3 x = 4cos 3 x – 3cos x

39. Hence find ∫ cos 3 x dx cos 3x= 4cos 3 x – 3cos x

41. The curve y = f(x) passes through the point ( π / 12, 1). f ’ (x) = cos 2x. Find f(x).