This document is a summary of key points from a mathematics exam preparation workshop on differentiation: 1) It provides the solutions to sample differentiation problems, finding the first and second derivatives of functions like f(x)=5x^3 and p(x)=5x^3-3x^2+7/x^2. 2) It also gives the solutions to differentiation questions that were part of past mathematics exams, finding derivatives like dy/dx of functions such as y=(x^3)(4x^2-2x). 3) The document aims to help students preparing for exams by providing worked examples of common differentiation questions and techniques for finding derivatives.

1. BBMP 1103
Mathematic Management
Exam Preparation Workshop Sept 2011
Part 2 - Differentiation
Presented By: Dr Richard Ng
26 Nov 2011
2ptg – 4ptg

2. 2. Focus on Differentiation
Question: 4 (May 2010)
Differentiate the following functions:
a) f ( x) 5 x3 Find f ' ( x) and f ' ' ( x) 3 marks
2
b) f ( x) x 3 (x6 2 x 3 1) Find f ' ( x) 3 marks
c) f ( x) [( x 3)(4 x 2 2 x)] Find f ' ' ( x) 4 marks
5 x 3 3x 2 7
d) p( x) Find p' ' ( x) 3 marks
x2
Prepared by Dr Richard Ng (2011) Page 2

3. Suggested Answers:
3 1
a) f ( x) 5 x 15 2
3
f' x
2 2
f ( x) 5x
1
1 15 2
1 f '' x
3 2 2 2
f' (5) x
2
1
1
15 15 2
f' x
2
f '' x
2 4
Prepared by Dr Richard Ng (2011) Page 3

4. 2
3
b) f ( x) x (x6 2 x 3 1)
2 1
3 5 2 6 3 2 3
f' x 6x 6x (x 2x 1) x
3
c) f ( x) x 3 (4 x 2 2 x)
f' x 3 8x 2 (4 x 2 2 x) 1
f '' x 3 8 (8x 2) 1 (8x 2)
8x 24 (8x 2) (8x 2)
24x 28
Prepared by Dr Richard Ng (2011) Page 4

5. 5 x 3 3x 2 7
d) p ( x)
x2
5x 3 3x 2 7
x2 x2 x2
2
5x 3 7 x
3
p' 5 ( 2)7 x
3
5 14x
4
p' ' 42x
Prepared by Dr Richard Ng (2011) Page 5

6. Question: 5 (January 2011)

7. a) y 12x 3 442x 2 300x 6
3x 1
d) y
dy 5x 2
36x 2 884x 300
dx
dy (5 x 2)[3] (3x 1)[5]
dx (5 x 2) 2
1 2
b) y x
x2 15x 6 (15x 5)
dy
2x 3 (5 x 2) 2
dx
5 11
x 3x 1 5 1
c) y x x (5 x 2) 2
15 15 5
dy 5 4 1 1 4 1
x x
dx 15 5 3 5

8. x 2 2x 1
e) y
x2 1
x2 2x 1 ( x 1)( x 1) ( x 1)
x2 1 ( x 1)( x 1) ( x 1)
dy ( x 1)[1] ( x 1)[1]
dx ( x 1) 2
x 1 x 1
( x 1) 2
2
( x 1) 2

9. Question: 6 (January 2010)

10. End of
Part 2