1. Derivatives – Outcomes
Find first derivatives of linear, quadratic, and cubic functions.
Find second derivatives of linear, quadratic, and cubic functions.
Solve problems about slopes and tangent lines.
Solve problems about maxima and minima.
Solve problems about rates of change.
Solve context problems.
Sketch curves using derivatives.

2. Find First Derivatives
For a function 𝑓 𝑥 , it is useful to know not just its value, but also how its value changes with 𝑥.
For linear functions, 𝑓 𝑥 =𝑎𝑥+𝑏, we have already studied slope, which provides this.
For more complicated functions, we use differentiation.
Differentiation is an operation which returns the derivative, 𝑓′(𝑥), of a function – this describes how 𝑓(𝑥) is changing around 𝑥, i.e. its rate of change.

3. Find First Derivatives
In words, “multiply by the current power, then reduce the power by 1”
For a simple polynomial function, 𝑓 𝑥 = 𝑥 𝑛 , the power rule states that its derivative, 𝑓 ′ 𝑥 =𝑛 𝑥 𝑛−1 .
e.g. 𝑓 𝑥 = 𝑥 2 ⇒ 𝑓 ′ 𝑥 =2𝑥
e.g. 𝑓 𝑥 = 𝑥 3 ⇒ 𝑓 ′ 𝑥 =3 𝑥 2
We sometimes use a different notation. For 𝑦= 𝑥 𝑛 , its derivative is d𝑦 d𝑥 =𝑛 𝑥 𝑛−1
e.g. 𝑦= 𝑥 2 ⇒ d𝑦 d𝑥 =2𝑥
e.g. 𝑦= 𝑥 3 ⇒ d𝑦 d𝑥 =3 𝑥 2

4. Find First Derivatives
𝑓 𝑥 = 𝑥 𝑛
⇒ 𝑓 ′ 𝑥 =𝑛 𝑥 𝑛−1
e.g. Find the derivative of each of the following functions:
𝑓 𝑥 = 𝑥 2
𝑦= 𝑥 3
𝑓 𝑥 =𝑥
𝑦=6
𝑦= 𝑥 𝑛
⇒ d𝑦 d𝑥 =𝑛 𝑥 𝑛−1

5. Find First Derivatives
For more complicated polynomial functions, we need two additional rules:
𝑦=𝑐𝑓 𝑥 ⇒ 𝑦 ′ =𝑐 𝑓 ′ 𝑥
i.e. coefficients of a function remain in product with the derivative of the function.
e.g. 𝑓 𝑥 =6 𝑥 3 ⇒ 𝑓 ′ 𝑥 =6 3 𝑥 2 =18 𝑥 2
𝑓 𝑥 =𝑢 𝑥 ±𝑣 𝑥 ±…⇒ 𝑓 ′ 𝑥 = 𝑢 ′ 𝑥 ± 𝑣 ′ 𝑥 ±…
i.e. terms of a function may be differentiated separately.
e.g. 𝑓 𝑥 = 𝑥 3 + 𝑥 2 +7⇒ 𝑓 ′ 𝑥 =3 𝑥 2 +2𝑥

6. Find First Derivatives
𝑓 𝑥 = 𝑥 𝑛
⇒ 𝑓 ′ 𝑥 =𝑛 𝑥 𝑛−1
e.g. Find the first derivative of each of the following:
𝑦=6𝑥+10
𝑓 𝑥 =3 𝑥 2 +5𝑥
𝑦= 𝑥 2 +2𝑥+7
𝑓(𝑥)=3 𝑥 3 −4 𝑥 2 −3𝑥+12
𝑦=1−𝑥− 𝑥 2
𝑓 𝑥 =−3 𝑥 3 +2 𝑥 2 −4𝑥+4
𝑦=(𝑥−1)(𝑥+1)
𝑠 𝑡 =18𝑡−2 𝑡 2
𝑣 𝑡 =96+40𝑡−4 𝑡 2
𝑓 𝑥 =5 𝑥 2 −8𝑥+ 𝑧 2
𝑦= 𝑥 𝑛
⇒ d𝑦 d𝑥 =𝑛 𝑥 𝑛−1

7. Find First Derivatives
2011(S) OL P1 Q8
Differentiate 𝑥 2 −6𝑥+1 with respect to 𝑥.
2006 OL P1 Q7
Differentiate 5 𝑥 3 −4𝑥+7 with respect to 𝑥.
2007 OL P1 Q7
Differentiate 6 𝑥 4 −3 𝑥 2 +7𝑥 with respect to 𝑥.
2007 OL P1 Q6
Let 𝑔 𝑥 = 𝑥 2 −6𝑥 𝑥∈ℝ
Write down 𝑔 ′ 𝑥 , the derivative of 𝑔 𝑥 .
For what value of 𝑥 is 𝑔 ′ 𝑥 =0?

8. Find Second Derivatives
Differentiating a first derivative gives the second derivative.
It works exactly the same way as finding the first derivative, though the notation is slightly different:
𝑓 𝑥 = 𝑥 2 ⇒ 𝑓 ′ 𝑥 =2𝑥⇒ 𝑓 ′′ 𝑥 =2
𝑦= 𝑥 3 ⇒ d𝑦 d𝑥 =3 𝑥 2 ⇒ d 2 𝑦 d 𝑥 2 =6𝑥
You always have to find the first derivative before finding the second derivative.

9. Find Second Derivatives
𝑓 𝑥 = 𝑥 𝑛
⇒ 𝑓 ′ 𝑥 =𝑛 𝑥 𝑛−1
e.g. Find the second derivative of each of the following:
𝑦=6𝑥+10
𝑓 𝑥 =3 𝑥 2 +5𝑥
𝑦= 𝑥 2 +2𝑥+7
𝑓(𝑥)=3 𝑥 3 −4 𝑥 2 −3𝑥+12
𝑦=1−𝑥− 𝑥 2
𝑓 𝑥 =−3 𝑥 3 +2 𝑥 2 −4𝑥+4
𝑦=(𝑥−1)(𝑥+1)
𝑠 𝑡 =18𝑡−2 𝑡 2
𝑣 𝑡 =96+40𝑡−4 𝑡 2
𝑦= 𝑥 𝑛
⇒ d𝑦 d𝑥 =𝑛 𝑥 𝑛−1

10. Solve Problems about Slopes and Tangents
Graphically, the derivative of a function represents the slope of a tangent to the function.
i.e. for a function, 𝑓 𝑥 , its derivative, 𝑓 ′ 𝑥 , is the slope of the function.
The derivative itself is also a function so its value changes with the value of 𝑥.
e.g. Find the slope of the tangent to 𝑓 𝑥 = 𝑥 2 −1 at the point 3,8 .
𝑓 ′ 𝑥 =2𝑥
𝑓 ′ 3 =2×3=6

11. Solve Problems about Slopes and Tangents
e.g. Let 𝑓 𝑥 = 𝑥 2 −4𝑥−5
Find the slope of the tangent to 𝑓 𝑥 at 1, −8 .
Find the coordinates of the point at which the slope of the tangent is 10.
𝑓 ′ 𝑥 =2𝑥−4
𝑓 ′ 1 =2−4=−2
𝑓 ′ 𝑥 =2𝑥−4=10⇒𝑥=7
𝑓 7 = 7 2 −4 7 −5=16
∴ 7,16 is where the slope of the tangent is 10.
Remember:
𝑓 𝑥 relates 𝑥 and 𝑦 coordinates to each other.
𝑓 ′ 𝑥 relates 𝑥 to the slope of 𝑓 𝑥 .

12. Solve Problems about Slopes and Tangents
e.g Find the equation of the tangent to 𝑓 𝑥 = 𝑥 2 +3𝑥 at 1,2 .
Equation requires point and slope: 𝑦− 𝑦 1 =𝑚 𝑥− 𝑥 1
1,2 works as point, need slope of tangent.
𝑓 ′ 𝑥 =2𝑥+3
𝑓 ′ 1 =2 1 +3=5
𝑦−2=5(𝑥−1)
⇒𝑦=5𝑥−3

13. Solve Problems about Slopes and Tangents
Find the slope of the tangent to each of the following curves at the given point:
𝑓 𝑥 =3−2𝑥 −1,5
𝑓 𝑥 = 𝑥 −3,10
𝑓 𝑥 =5 𝑥 , 13
𝑓 𝑥 =28 𝑥 ,28
𝑓 𝑥 = 𝑥 2 +2𝑥+5 −1,4
𝑓 𝑥 = 𝑥 3 + 𝑥 ,0

14. Solve Problems about Slopes and Tangents
Find the equation of the tangent to each of the following curves at the given value of 𝑥:
𝑦= 𝑥 𝑥=2
𝑦= 𝑥 𝑥=−1
𝑦= 𝑥 3 −1 𝑥=−2
𝑦= 𝑥 3 +2𝑥 𝑥=1
𝑦= 𝑥 3 −3 𝑥 𝑥=3
𝑦= 𝑥 3 −2 𝑥 𝑥=2
𝑦− 𝑦 1 =𝑚 𝑥− 𝑥 1

15. Solve Problems about Slopes and Tangents
2003 OL P1 Q8
Let 𝑓 𝑥 = 𝑥 3 +2 𝑥 2 −1
Find 𝑓 ′ 𝑥 , the derivative of 𝑓 𝑥 .
𝐿 is the tangent to the curve 𝑦=𝑓 𝑥 at 𝑥=− Find the slope of 𝐿.
Find the two values of 𝑥 at which the tangents to the curve 𝑦=𝑓 𝑥 are perpendicular to 𝐿.
Recall that perpendicular lines have reciprocal negative slopes
i.e. if 𝑚 1 = 𝑎 𝑏 , 𝑚 2 =− 𝑏 𝑎

16. Solve Problems about Slopes and Tangents
2012 (O)OL P1 Q8
Let ℎ 𝑥 =5+3𝑥− 𝑥 2 , where 𝑥∈ℝ.
Find the coordinates of the point 𝑃 at which the curve 𝑦=ℎ 𝑥 cuts the 𝑦-axis.
Find the equation of the tangent to the curve 𝑦=ℎ 𝑥 at 𝑃.
The tangent to the curve 𝑦=ℎ 𝑥 at 𝑥=𝑡 is perpendicular to the tangent at 𝑃. Find the value of 𝑡.
𝑦− 𝑦 1 =𝑚 𝑥− 𝑥 1

17. Solve Problems about Maxima and Minima
Important points for curves are turning points / stationary points.
They come in two types: maximum and minimum.
We often refer to “local” maximum and minimum to distinguish when there are more than one of either

18. Solve Problems about Maxima and Minima
If the derivative of a function represents the slope of the tangent, what value does the derivative take at turning points?

19. Solve Problems about Maxima and Minima
e.g. Find the coordinates of the maximum and minimum points of 𝑓 𝑥 = 𝑥 3 −9 𝑥 2 +24𝑥−10
For turning points, d𝑦 d𝑥 =0
⇒3 𝑥 2 −18𝑥+24=0
⇒ 𝑥 2 −6𝑥+8=0
⇒ 𝑥−4 𝑥−2 =0
⇒𝑥=4
𝑓 4 = 4 3 − −10 =6
4,6 - minimum
⇒𝑥=2
𝑓 2 = 2 3 − −10
=10
2,10 - maximum
How do you know which is maximum and minimum?

20. Solve Problems about Maxima and Minima
Know your curve:
𝑥 3 −9 𝑥 2 +24𝑥−10 is a positive cubic so looks like
The maximum has the higher 𝑦- coordinate and the minimum has the lower 𝑦-coordinate.
For a positive cubic (+ 𝑥 3 ), the maximum is to the left of the minimum.

21. Solve Problems about Maxima and Minima
Know your curve:
𝑓 𝑥 =− 𝑥 3 +2𝑥
The maximum has the higher 𝑦- coordinate and the minimum has the lower 𝑦-coordinate.
For a negative cubic (− 𝑥 3 ), the maximum is to the right of the minimum.

22. Solve Problems about Maxima and Minima
Know your curve:
Positive quadratics (+ 𝑥 2 ) have a minimum.
Negative quadratics (− 𝑥 2 ) have a minimum.

23. Solve Problems about Maxima and Minima
Alternatively, use the second derivative test:
For minima, d 2 𝑦 d 𝑥 2 >0
For maxima, d 2 𝑦 d 𝑥 2 <0
For 𝑦= 𝑥 3 −9 𝑥 2 +24𝑥−10
d 2 𝑦 d 𝑥 2 =6𝑥−18 for this function.
d 2 𝑦 d 𝑥 =6 4 −18=6>0⇒ minimum
d 2 𝑦 d 𝑥 =6 2 −18=−6<0⇒ maximum

24. Solve Problems about Maxima and Minima
Find the turning point(s) of each of the following curves and state whether they are a maximum or minimum:
𝑦= 𝑥 2 +4𝑥+1
𝑦=6𝑥−2 𝑥 2
𝑦=−4 𝑥 2 +4𝑥+1
𝑦=2 𝑥 3 +3 𝑥 2 −36𝑥
𝑦= 𝑥 𝑥 2 +36𝑥+6
𝑦=− 𝑥 3 +6 𝑥 2 +96𝑥−12

25. Solve Problems about Maxima and Minima
2006 OL P1 Q6
Let 𝑓 𝑥 =3+8𝑥−2 𝑥 2 𝑥∈ℝ
Find the coordinates of the point at which the curve 𝑦=𝑓 𝑥 cuts the 𝑦-axis.
Find the value of 𝑥 for which 𝑓 𝑥 is a maximum.
For what range of values of 𝑥 is 𝑓 ′ 𝑥 >4?

26. Solve Problems about Maxima and Minima
2005 OL P1 Q6
Let 𝑓 𝑥 = 𝑥 2 +𝑝𝑥+10, 𝑥∈ℝ, 𝑝∈ℤ
Find 𝑓 ′ 𝑥 , the derivative of 𝑓 𝑥 .
The minimum value of 𝑓 𝑥 is at 𝑥=3. Find the value of 𝑝.
Find the equation of the tangent to 𝑓 𝑥 at the point 0, 10 .
𝑦− 𝑦 1 =𝑚 𝑥− 𝑥 1

27. Solve Problems about Rates of Change
Aside from graphs, derivatives represent rates of change.
In general for 𝑓 𝑥 :
𝑓 ′ 𝑥 =0 indicates that 𝑓 𝑥 is not changing (and recall that it also indicates a maximum or minimum).
𝑓 ′ 𝑥 <0 indicates that 𝑓 𝑥 is decreasing.
𝑓 ′ 𝑥 >0 indicates that 𝑓 𝑥 is increasing.

28. Solve Problems about Rates of Change
e.g. find the values of 𝑥 for which 𝑓 𝑥 = 𝑥 2 −8𝑥+5 is increasing and for which it is decreasing.
Check for turning points:
𝑓 ′ 𝑥 =2𝑥−8=0⇒𝑥=4
Check 𝑓 ′ 𝑥 either side of this value.
[𝑥<4] 𝑓 ′ 0 =2 0 −8=−8<0
Thus 𝑓 𝑥 is decreasing for 𝑥<4.
[𝑥>4] 𝑓 ′ 5 =2 5 −8=2>0
Thus 𝑓 𝑥 is increasing for 𝑥>4.
ANY number less than 4 is fine: 0 is easy!

29. Solve Problems about Rates of Change
e.g. State the range of values of 𝑥 (or 𝑡) for which each of the following are increasing and decreasing.
𝑓 𝑥 =3 𝑥 2 +5𝑥
𝑦= 𝑥 2 +2𝑥+7
𝑓(𝑥)=3 𝑥 3 −4 𝑥 2 −3𝑥+12
𝑦=1−𝑥− 𝑥 2
𝑓 𝑥 =−3 𝑥 3 +2 𝑥 2 −4𝑥+4
𝑦=(𝑥−1)(𝑥+1)
𝑠 𝑡 =18𝑡−2 𝑡 2
𝑣 𝑡 =96+40𝑡−4 𝑡 2

30. Solve Context Problems
For many word problems, real-world terms are used in place of mathematical ones.
e.g. instead of maximum, you may be asked about the largest, biggest, longest, tallest, most, heaviest, quickest, strongest, etc.
e.g. instead of minimum, you may be asked about the smallest, shortest, least, lightest, slowest, weakest, etc.
Likewise, instead of increasing/decreasing, you will see bigger/smaller, taller/shorter, heavier/lighter etc.

31. Solve Context Problems
2012 OL P1 Q9
Investments can increase or decrease in value. The value of a particular investment of €100 was found to fit the following model:
𝑉=100+45𝑡−1.5 𝑡 2
, where 𝑉 is the value of the investment in euro and 𝑡 is the time in months after the investment was made.
Find the rate at which the value of the investment was changing after 6 months.
State whether the value of the investment was increasing or decreasing after 18 months. Justify your answer.

32. Solve Context Problems
2012 OL P1 Q9
[continued]
Investments can increase or decrease in value. The value of a particular investment of €100 was found to fit the following model:
𝑉=100+45𝑡−1.5 𝑡 2
The investment was cashed in at the end of 24 months. How much was it worth at that time?
How much was the investment worth when it had its maximum value?

33. Solve Context Problems
2012 (S) OL P1 Q9
A farmer is growing winter wheat. The amount of wheat he will get per hectare depends on, among other things, the amount of nitrogen fertiliser that he uses. For his particular farm, the amount of wheat depends on the nitrogen in the following way:
𝑌= 𝑁−0.1 𝑁 2
where 𝑌 is the amount of wheat produced, in kg per hectare, and 𝑁 is the amount of nitrogen added, in kg per hectare.
How much wheat will he get per hectare if he uses 100 kg of nitrogen per hectare?

34. Solve Context Problems
2012 (S) OL P1 Q9
[continued]
Find the amount of nitrogen that he must use in order to maximise the amount of wheat produced.
What is the maximum possible amount of wheat produced per hectare?
The farmer’s total costs for producing the wheat are €1300 per hectare. He can sell the wheat for €160 per tonne. He can also get €75 per hectare for the leftover straw. If he achieves the maximum amount of wheat, what is his profit per hectare?

35. Solve Context Problems
Distance-time problems come up a lot and you will need to remember the notation and terms used for functions and derivatives in this context:
Displacement, 𝑠 𝑡 , refers to distance in a direction.
Velocity, 𝑣 𝑡 , refers to speed in a direction. 𝑣 𝑡 = 𝑠 ′ 𝑡 .
Acceleration, 𝑎 𝑡 , is the first derivative of velocity and the second derivative of displacement: 𝑎 𝑡 = 𝑣 ′ 𝑡 = 𝑠 ′′ 𝑡 .
Usually positive values are in one direction and negative values in the other direction.
Sometimes ℎ 𝑡 is used as height instead of displacement
“at rest” means velocity is zero

36. Solve Context Problems
2003 OL P1 Q7
A train is driving along a track. Suddenly, the brakes are applied. From the time the brakes are applied (𝑡=0 seconds), the distance travelled by the train, in metres, is given by:
𝑠=30𝑡− 1 4 𝑡 2
What is the speed of the train at the moment the brakes are applied?
How many seconds does it take for the train to come to rest?
How far does the train travel in that time?

37. Solve Context Problems
2004 OL P1 Q7
A jet is moving along an airport runway. When it passes a marker it begins to accelerate for take-off. From the time the jet passes the marker, its distance from the marker is given by
𝑠=2 𝑡 2 +3𝑡, where 𝑠 is in metres and 𝑡 is in seconds.
Find the speed of the jet at the instant it passes the marker (𝑡=0).
The jet has to reach a speed of 83 metres per second to take off. After how many seconds will the jet reach this speed?
How far is the jet from the marker at that time?
Find the acceleration of the jet.

38. Solve Context Problems
2006 OL P1 Q7
A missile is fired straight up in the air. The height, ℎ metres, of the missile above the firing position is given by
ℎ=𝑡 200−5𝑡
Where 𝑡 is the time in seconds from the instant the missile fired.
Find the speed of the missile after 10 seconds.
Find the acceleration of the missile.
One second before reaching its greatest possible height, the missile strikes a target. Find the height of the target.

39. Solve Context Problems
Area and volume problems are also very common.
You often have to create a formula for area or volume from given measurements.
e.g. A new rectangular garden is being built onto the side of a building. 600 m of fencing is to be used to close off the other three sides. Find its maximum area.

40. Solve Context Problems
Draw diagram.

41. Solve Context Problems
Form equation.

42. Solve Context Problems
Set derivative to 0 and solve.

43. Solve Context Problems
2012 OL P1 Q9
Garden paving slabs measure 40 cm by 20 cm. The slabs are to be arranged to form a rectangular paved area. There are 𝑥 slabs along one side and 𝑦 slabs along an adjacent side, as shown.
Write the length of the perimeter, in centimetres, in terms of 𝑥 and 𝑦.
The material being used for edging means that the perimeter is to be 64 metres. Find 𝑦 in terms of 𝑥.
Find the value of x for which the paved area is as large as possible.
Find the number of slabs needed to pave this maximum area.

44. Sketch Curves using Derivatives
Here is a graph of a cubic and its derivatives:
What is happening to the order of the polynomial as we differentiate?
Note how the turning points of one graph corresponds to a root of its derivative

45. Sketch Curves using Derivatives
Recall that when differentiating, the power of each term reduces by 1.
This means that functions reduce their order as we continue to differentiate them, i.e. they follow:
cubic → quadratic → linear.
When drawing curves and their derivatives, you should have no more than one of any shape.

46. Sketch Curves using Derivatives
Turning points on one curve are roots on their derivatives:

47. Sketch Curves using Derivatives
For each of the following curves, (i) find their turning points, (ii) find where they are increasing and decreasing, and (iii) sketch 𝑓 𝑥 .
𝑦= 𝑥 2 +4𝑥+1
𝑦=6𝑥−2 𝑥 2
𝑦=−4 𝑥 2 +4𝑥+1
𝑦=2 𝑥 3 +3 𝑥 2 −36𝑥
𝑦= 𝑥 𝑥 2 +36𝑥+6
𝑦=− 𝑥 3 +6 𝑥 2 +96𝑥−12

48. Sketch Curves using Derivatives
2003 OL P1 Q6
Let 𝑓 𝑥 =3−5𝑥−2 𝑥 2
Find 𝑓 ′ 𝑥 , the derivative of 𝑓 𝑥 , and hence find the co-ordinates of the local maximum point of the curve 𝑦=𝑓 𝑥 .
Solve the equation 𝑓 𝑥 =0.
Use your answers from parts (i) and (ii) to sketch the graph of 𝑓:𝑥→3−5𝑥−2 𝑥 2 , showing scaled and labelled axes.

49. Sketch Curves using Derivatives
2008 OL P1 Q8
Let 𝑓 𝑥 = 𝑥 3 −9 𝑥 2 +24𝑥−18, where 𝑥∈ℝ.
Find 𝑓 1 and 𝑓 5 .
Find 𝑓 ′ 𝑥 , the derivative of 𝑓 𝑥 .
Find the coordinates of the local maximum point and of the local minimum point of the curve 𝑦=𝑓 𝑥 .
Draw the graph of the function 𝑓 in the domain 1≤𝑥≤ 5.
Use your graph to write down the range of values of 𝑥 for which 𝑓 ′ 𝑥 <0.
The line 𝑦=−3𝑥+𝑐 is a tangent to the curve 𝑦=𝑓 𝑥 . Find the value of 𝑐.