1. Differentiation with Trig – Outcomes
Differentiate sin 𝑥 , cos 𝑥 , and tan 𝑥 , as well as sums, real multiples, composites, products, and quotients with these functions.
Find higher derivatives of trigonometric functions, particularly second derivatives.
Solve problems about gradients and rates of change.
Solve problems about tangents and normals to curves.
Solve problems about maxima and minima.
Solve problems about inflexion.
Discuss the behaviour of graphs and the relationship between the graphs of 𝑓 𝑥 , 𝑓 ′ 𝑥 , and 𝑓 ′′ 𝑥 .

2. Differentiation with Trig – Outcomes
Solve optimization problems.
Solve kinematics problems.

3. Differentiate sin 𝑥 , cos 𝑥 , and tan 𝑥
Three trigonometric functions must be differentiated on our course:
𝑓 𝑥 = sin 𝑥 ⇒ 𝑓 ′ 𝑥 = cos 𝑥
𝑓 𝑥 = cos 𝑥 ⇒ 𝑓 ′ 𝑥 =− sin 𝑥
𝑓 𝑥 = tan 𝑥 ⇒ 𝑓 ′ 𝑥 = 1 cos 2 𝑥

4. Differentiate sin 𝑥 , cos 𝑥 , and tan 𝑥
sine sim
Recall from trigonometry:
𝑓 𝑥 =𝑎 sin 𝑏(𝑥+𝑐) +𝑑 (likewise for cos and tan), where:
𝑎 is the amplitude of the function – its max deviation from the mean position.
𝑏 affects the period of the function – how long it takes to repeat itself. Specifically the period is 2𝜋 𝑏 . ( 𝜋 𝑏 for tan )
𝑐 is the horizontal translation/shift/offset of the function. Positive value means left shift, negative value means right shift.
𝑑 is the vertical translation of the function. Positive value means up shift, negative value means down shift.

5. Solve Problems about Composite Functions
Chain Rule:
𝑦=𝑢 𝑣 𝑥
⇒ d𝑦 d𝑥 = d𝑦 d𝑢 × d𝑢 d𝑥
e.g. Differentiate each of the following with respect to 𝑥:
𝑓 𝑥 = sin 2𝑥
𝑓 𝑥 = cos 2𝑥
𝑓 𝑥 = tan 2𝑥
𝑦= sin 2 𝑥
𝑦= cos 2 𝑥
𝑦= tan 2 𝑥
𝑦= sin 3 2𝑥
𝑓 𝑥 = cos 2 𝑥 2 +7
𝑦= tan 4 (2 𝑥 3 +4 𝑥 2 −6𝑥−8)

6. Solve Problems about Products
Product Rule:
𝑦=𝑢𝑣
⇒ d𝑦 d𝑥 =𝑢 d𝑣 d𝑥 +𝑣 d𝑢 d𝑥
e.g. Differentiate each of the following with respect to 𝑥:
𝑓 𝑥 =𝑥 sin 𝑥
𝑓 𝑥 = 𝑥 2 cos 𝑥
𝑓 𝑥 = 𝑒 𝑥 tan 𝑥
𝑓 𝑥 = ln 𝑥 sin 𝑥
𝑓 𝑥 = 𝑥 2 +4𝑥+3 sin 𝑥
𝑦=2 cos 𝑥 sin 𝑥
𝑦= 𝑒 𝑥 ln 𝑥 sin 𝑥

7. Solve Problems about Quotients
Quotient Rule:
𝑦= 𝑢 𝑣
⇒ d𝑦 d𝑥 = 𝑣 d𝑢 d𝑥 −𝑢 d𝑣 d𝑥 𝑣 2
e.g. Differentiate each of the following with respect to 𝑥:
𝑓 𝑥 = sin 𝑥 𝑥
𝑓 𝑥 = 1 tan 𝑥
𝑓 𝑥 = 𝑥 cos 𝑥
𝑓 𝑥 = sin 𝑥 1+ cos 𝑥
𝑦= cos 2 𝑥 sin 2 𝑥
𝑦= sin 2𝑥 2+ cos 2𝑥
𝑦= sin 3𝑥 𝑒 𝑥

8. Find Higher Derivatives
Find the second derivative of each of the following:
𝑓 𝑥 = sin 𝑥
𝑓 𝑥 = cos 𝑥
𝑓 𝑥 = tan 𝑥
𝑓 𝑥 = sin 2𝑥
𝑓 𝑥 = cos 3𝑥
𝑓 𝑥 = sin 2 𝑥
𝑓 𝑥 = cos 3 𝑥
𝑓 𝑥 = tan 2 2𝑥

9. Solve Problems about Gradients
Let 𝑓 𝑥 = sin 2𝑥
Find the equations of the tangent and normal to the graph of 𝑓 𝑥 at 𝑥= 𝜋 3 .
𝑓 𝜋 3 = sin 2𝜋 3 =
𝑓 ′ 𝑥 =2 cos 2𝑥
𝑚 𝑇 = 𝑓 ′ 𝜋 3 =2 cos 2𝜋 3 =−1
⇒ 𝑚 𝑁 =1

10. Solve Problems about Gradients
𝑥 1 , 𝑦 1 = 𝜋 3 , ; 𝑚 𝑇 =−1; 𝑚 𝑁 =1
Tangent: 𝑦− =−1 𝑥− 𝜋 3
⇒𝑦=−𝑥+ 𝜋 =−𝑥+1.91
Normal: 𝑦− =1 𝑥− 𝜋 3
⇒𝑦=𝑥− 𝜋 =𝑥+0.252

11. Solve Problems about Gradients
QB T6P1 Q1
Let 𝑔 𝑥 =2𝑥 sin 𝑥
Find 𝑔 ′ 𝑥 .
Find the gradient of the graph of 𝑔 at 𝑥=𝜋.
QB T6P1 Q9
Let ℎ 𝑥 = 6𝑥 cos 𝑥
Find ℎ ′ 0 .
QB T6P4 Q20
Consider 𝑓 𝑥 = 𝑥 2 sin 𝑥
Find 𝑓 ′ 𝑥 .
Find the gradient of the curve of 𝑓 at 𝑥= 𝜋 2 .

12. Solve Problems about Gradients
QB T6P2 Q4
Let 𝑓 𝑥 = 𝑒 2𝑥 cos 𝑥 , −1≤𝑥≤2.
Show that 𝑓 ′ 𝑥 = 𝑒 2𝑥 2 cos 𝑥 − sin 𝑥 .
Let the line 𝐿 be normal to the curve of 𝑓 at 𝑥=0. Find the equation of 𝐿.
The graph of 𝑓 and the line 𝐿 intersect at the point 0, 1 and at a second point 𝑃. Find the 𝑥-coordinate of 𝑃.

13. Solve Problems about Gradients
11M.2.sl.TZ1.8
The following diagram shows a waterwheel with a bucket. The bucket rotates at a constant rate in an anticlockwise direction.
The diameter of the wheel is 8 metres. The centre of the wheel, 𝐴, is 2 metres above the water level. After 𝑡 seconds, the height of the bucket above the water level is given by ℎ=𝑎 sin 𝑏𝑡 +2.
Show that 𝑎=4.
The wheel turns at a rate of one rotation every 30 seconds. Show that 𝑏= 𝜋 15 .

14. Solve Problems about Gradients
The diameter of the wheel is 8 metres. The centre of the wheel, 𝐴, is 2 metres above the water level. After 𝑡 seconds, the height of the bucket above the water level is given by ℎ=4 sin 𝜋 15 𝑡 +2.
In the first rotation, there are two values of 𝑡 when the bucket is descending at a rate of 0.5 ms−1.
Find these values of 𝑡.
Determine whether the bucket is underwater at the second value of 𝑡.

15. Solve Problems about Extrema
Recall what sin and cos curves look like:

16. Solve Problems about Extrema
tan has no extrema, so we need not consider it.
sin and cos each have infinitely many extrema, so we often restrict our domain to 0 𝑜 ≤𝑥≤ 360 𝑜 (or in radians, 0≤𝑥≤2𝜋).
This may vary with argument (e.g. sin 2𝑥 may be restricted to 0 𝑜 ≤𝑥≤ 180 𝑜 instead) as a change in period may increase or decrease the number of extrema in a given domain.
e.g. Find the coordinates of the extrema of 𝑓 𝑥 = sin 𝑥 for 0≤𝑥≤2𝜋.

17. Solve Problems about Extrema
For extrema, 𝑓 ′ 𝑥 =0.
𝑓 ′ 𝑥 = cos 𝑥 =0
𝑥= 𝜋 2 , 3𝜋 2
𝑓 𝜋 2 = sin 𝜋 2 =1 (maximum)
𝜋 2 , 1
𝑓 3𝜋 2 = sin 3𝜋 2 =−1 (minimum)
3𝜋 2 ,−1

18. Solve Problems about Extrema
e.g. Find the 𝑥-coordinate of all extrema of each of the following functions for 0≤𝑥≤2𝜋.
cos 𝑥
sin 2𝑥
cos 3𝑥
sin 2𝑥+ 𝜋 2
cos 𝑥−𝜋
2 sin 4𝑥+2𝜋

19. Solve Problems about Extrema
QB T6P3 Q21
The following diagram shows the graph of 𝑓 𝑥 =𝑎 sin (𝑏 𝑥−𝑐 ) +𝑑, for 2≤𝑥≤10.
There is a maximum at 𝑃(4, 12) and a minimum at 𝑄(8, 14).
Use the graph to write down the value of 𝑎, 𝑐, and 𝑑.
Show that 𝑏= 𝜋 4 .
Find 𝑓 ′ 𝑥 .
At a point 𝑅, the gradient is −2𝜋. Find the 𝑥-coordinate of 𝑅.

20. Solve Problems about Points of Inflexion
Recall what sin and cos curves look like:

21. Solve Problems about Points of Inflexion
This time we should consider tan curves also:

22. Solve Problems about Points of Inflexion
e.g. Find the 𝑥-coordinates of the points of inflexion of sin 𝑥 in the domain 0≤𝑥≤2𝜋.
𝑓 ′ 𝑥 = cos 𝑥 ⇒ 𝑓 ′′ 𝑥 =− sin 𝑥
Second derivative is zero for points of inflexion:
⇒− sin 𝑥 =0
Which gives 𝑥=0, 𝜋, 2𝜋.
Now check the value of the second derivative around these 𝑥 values: 𝑥 𝜋 2𝜋
− sin 𝑥
+++
---
Thus, 𝑥=0,𝜋,2𝜋 are all locations of points of inflexion.

23. Solve Problems about Points of Inflexion
e.g. Find the 𝑥-coordinate of all points of inflexion of each of the following functions for 0≤𝑥≤2𝜋.
cos 𝑥
tan 𝑥
sin 2𝑥
cos 3𝑥
tan 2𝑥
sin 2𝑥+ 𝜋 2
cos 𝑥−𝜋
2 sin 4𝑥+2𝜋

24. Solve Problems about Points of Inflexion
10M.1.sl.TZ1.9
Let 𝑓 𝑥 = cos 𝑥 sin 𝑥 , for sin 𝑥 ≠0.
In the following table, 𝑓 ′ 𝜋 2 =𝑝 and 𝑓 ′′ 𝜋 2 =𝑞. It also gives approximate values of 𝑓 ′ 𝑥 and 𝑓 ′′ 𝑥 near 𝑥= 𝜋 2 .
Use the quotient rule to show that 𝑓 ′ 𝑥 =− 1 sin 2 𝑥 .
Find 𝑓 ′′ 𝑥 .
Find the values of 𝑝 and 𝑞.
Use the information in the table to explain why there is a point of inflexion on the graph of 𝑓 where 𝑥= 𝜋 2 . 𝑥
𝜋 2 −0.1
𝜋 2 𝜋
𝑓 ′ 𝑥
−1.01 𝑝
𝑓 ′′ 𝑥
0.203 𝑞
−0.203

25. Discuss 𝑓(𝑥), 𝑓 ′ 𝑥 , and 𝑓 ′′ 𝑥
Here is a graph of sin 𝑥 and its derivatives:

26. Discuss 𝑓(𝑥), 𝑓 ′ 𝑥 , and 𝑓 ′′ 𝑥
Here is a graph of cos 𝑥 and its derivatives:

27. Discuss 𝑓(𝑥), 𝑓 ′ 𝑥 , and 𝑓 ′′ 𝑥
Here is a graph of tan 𝑥 and its derivatives:

28. Solve Optimization Problems
11M.2.sl.TZ2.10
The diagram shows a plan for a window in the shape of a trapezium.
Three sides of the window are 2 m long. The angle between the sloping sides of the window and the bas is 𝜃, where 0≤𝜃≤ 𝜋 2 .
Show that the area of the window is given by 𝑦=4 sin 𝜃 +2 sin 2𝜃 .
Zoe wants a window to have an area of 5 m2. Find the two possible values of 𝜃.
John wants two windows which have the same area 𝐴 but different values of 𝜃. Find all possible values for 𝐴.

29. Solve Optimization Problems
08M.1.sl.TZ2.10
The following diagram shows a semicircle with centre 𝑂, diameter 𝐴𝐵 , with radius 2.
Let 𝑃 be a point on the circumference, with 𝑃 𝑂 𝐵=𝜃 radians. 𝑆 is the area of the two segments shaded below.

30. Solve Optimization Problems
Find the area of the triangle 𝑂𝑃𝐵 in terms of 𝜃.
Explain why the area of triangle 𝑂𝑃𝐴 is the same as the area of triangle 𝑂𝑃𝐵.
Show that 𝑆=2 𝜋−2 sin 𝜃 .
Find the value of 𝜃 when 𝑆 is a local minimum, justifying that it is a minimum.
Find a value of 𝜃 for which 𝑆 has its greatest value.

31. Solve Kinematic Problems
12N.2.sl.TZ0.7
A particle’s displacement, in metres, is given by 𝑠 𝑡 = 2𝑡 cos 𝑡 , for 0≤𝑡≤6, where 𝑡 is the time in seconds.
Sketch the graph of 𝑠.
Find the maximum velocity of the particle.

32. Solve Kinematic Problems
12M.1.sl.TZ1.10
Given that cos 𝜋 3 = 1 2 , and sin 𝜋 3 =
The displacement of an object from a fixed point, 𝑂, is given by 𝑠 𝑡 =𝑡− sin 2𝑡 for 0≤𝑡≤𝜋.
Find 𝑠 ′ 𝑡 .
In this interval, there are only two values of 𝑡 for which the object is not moving. One value is 𝑡= 𝜋 6 .
Find the other value.
Show that 𝑠′(𝑡)>0 between these two values of 𝑡.