1. “Keep the first, differentiate the second”
The Product Rule
In words:
“Keep the first, differentiate the second”
“Keep the second, differentiate the first” +

2. Examples: 1.
Differentiate

3. Examples: 2.
Differentiate
Now watch this.

4. Examples: 3.
Differentiate
Try this using “words”

6. The Quotient Rule In words:
“Keep the denominator, differentiate the numerator”
“Keep the numerator, differentiate the denominator” –
Denominator 2

7. Examples: 1.
Differentiate

8. Examples: 2.
Differentiate
Try this using “words”

9. Add a denominator here

10. Derivatives of New Functions
Definitions:
Reminder:
continue

11. p/2 p
3p/2 2p
-10 -5 5 10

12. p/2 p
3p/2 2p
-10 -5 5 10

13. p/2 p
3p/2 2p
-10 -5 5 10

14. Use the Quotient Rule now
Derivative of
Proof:
Use the Quotient Rule now

15. Prove these and keep with your notes.
Derivatives of
Prove these and keep with your notes.
Use chain rule or quotient rule

16. Example:
Given that
show that

17. Exponential and Logarithmic Functions
Reminder:
and
are inverse to each other.
They are perhaps the most important functions in the applications of calculus in the real world.
Alternative notation:
Two very useful results:
Learn these!
Also:
Practise changing from exp to log and vice-versa.

18. -6 -4 -2 2 4 6 -1 1 x y

19. Derivatives of the Exponential and Logarithmic Functions
(ii)
Proof of (ii)

20. Examples: 1.
Differentiate
Use the Chain Rule 2.
Differentiate
Use the Product Rule

21. 3.
Differentiate
Use the Chain Rule 4.
Differentiate
Use the Quotient Rule

22. Note:
In general
Useful for reverse i.e. INTEGRATION

23. Higher Derivatives
Given that f is differentiable, if is also differentiable then its derivative is denoted by
The two notations are:
function
1st derivative
2nd derivative ……
nth derivative f

24. Example:
If , write down is first second and third derivatives and hence make a conjecture about its nth derivative.
Conjecture:
The nth derivative is

25. Rectilinear Motion
If displacement from the origin is a function of time I.e.
then
v - velocity
a - acceleration

26. Example:
A body is moving in a straight line, so that after t seconds its displacement x metres from a fixed point O, is given by
(a)
Find the initial dislacement, velocity and acceleration of the body.
(b)
Find the time at which the body is instantaneously at rest.

27. Extreme Values of a Function
Understand the following terms:
Critical Points
Local Extreme Values
Local maximum
Local minimum
End Point Extreme Values
End Point maximum
End Point minimum
See, MIA Mathematics 1, Pages 54 – 55

28. The Nature of Stationary Points
Consider a curve and the corresponding gradient function A
Consider maximum turning point A.
Notice, gradient of for x in the neighbourhood of A is negative.
i.e is negative
Similarly, gradient of for x in the neighbourhood of B is positive. B
i.e is positive

29. The Nature of Stationary Points
Rule for Stationary Points
and minimum turning point
and maximum turning point
and possibly a point of inflexion but must check using a table of signs

30. Example:
Now what does look like?
Consider
At S.P.
Notice no Point of Inflexion.

31. See, MIA Mathematics 1, Pages 58 – 59
Global Extreme Values
Understand the following terms:
Global Extreme Values
Global maximum
Global minimum
See, MIA Mathematics 1, Pages 58 – 59

32. Find the coordinates and nature of the stationary point on the curve
Example:
Find the coordinates and nature of the stationary point on the curve
What does this curve look like?
At S.P.
is a Minimum Turning Point

33. x y

34. Optimisation Problems
A sector of a circle with radius r cm has an area of 16 cm2.
(a)
Show that the perimeter P cm of the sector is given by l
(b)
Find the minimum value of P. r
(a) r
now

35. (b)
At SP
r = 4 gives a minimum stationary value of