Maths
Chapter 22 Differen�a�on
In this chapter, you will learn about:
22.1 Differen�a�on
22.2 Gradient of a curve
22.3 Turning points.
22.1 Differen�a�on
𝒅𝒅𝒅𝒅
What is differen�a�on � �?
𝒅𝒅𝒅𝒅
It is to differen�ate the func�on y with respect to x. We know the func�on y is in terms of x. Thus,
𝒅𝒅𝒅𝒅
differen�a�on � � is the gradient func�on of the func�on y.
𝒅𝒅𝒅𝒅
Says a func�on of 𝒚𝒚 = 𝒙𝒙𝟐𝟐 + 𝟑𝟑𝟑𝟑 + 𝟓𝟓, then differen�ate this func�on and we get
𝒅𝒅𝒅𝒅
𝒅𝒅𝒅𝒅
How to perform differen�a�on?
𝑦𝑦 = 𝑥𝑥 𝑛𝑛
𝒚𝒚
𝑑𝑑𝑑𝑑
= 𝑛𝑛𝑥𝑥 𝑛𝑛−1
𝑑𝑑𝑑𝑑
𝑐𝑐
0
𝑥𝑥
1
𝑥𝑥 2
2𝑥𝑥
𝑥𝑥 3
𝑎𝑎𝑎𝑎 2 + 𝑏𝑏𝑏𝑏 + 𝑐𝑐
Ques�ons:
1. Find
𝒅𝒅𝒅𝒅
𝒅𝒅𝒅𝒅
𝒅𝒅𝒅𝒅
𝒅𝒅𝒅𝒅
3𝑥𝑥 2
2𝑎𝑎𝑎𝑎 + 𝑏𝑏
for each of following:
a) 𝑦𝑦 = 𝑥𝑥 2 + 6𝑥𝑥 + 8
c) 𝑦𝑦 = 𝑥𝑥 3 + 8
b) 𝑦𝑦 = 130
d) 𝑦𝑦 = (𝑥𝑥 + 2)(𝑥𝑥 + 5)
= 𝟐𝟐𝟐𝟐 + 𝟑𝟑.
Maths
Chapter 22 Differen�a�on
e) 𝑦𝑦 = 6𝑥𝑥 + 3𝑥𝑥 2
𝑦𝑦 = −𝑥𝑥 3 + 3𝑥𝑥 2
f)
g) 𝑦𝑦 = 0.2𝑥𝑥 2 − 6𝑥𝑥 + 0.6
h) 𝑦𝑦 = −6𝑥𝑥 + 5𝑥𝑥 3
22.2 Gradient of a curve
What is gradient of a curve?
It is tangent line to the curve at any point. Each point has a different/instantaneous gradient on the
curve.
How to find the gradient of a curve?
𝒅𝒅𝒅𝒅
𝒅𝒅𝒅𝒅
Differen�ate the func�on (𝑦𝑦) to get gradient func�on � � and subs�tute any point to get
instantaneous gradient of that point.
Ques�ons:
1. Find the gradient for each of following:
a) A curve has the equa�on 𝑦𝑦 = 0.5𝑥𝑥 2 + 4𝑥𝑥 − 3
𝑑𝑑𝑑𝑑
i.
Find
ii.
Find the gradient at (0, -3)
𝑑𝑑𝑑𝑑
iii.
Find the gradient at (2, 7)
iv.
Find the coordinate of the point
where the gradient is 5.
Maths
Chapter 22 Differen�a�on
b) A curve has the equa�on 𝑦𝑦 = 𝑥𝑥 2 − 6𝑥𝑥 + 15
𝑑𝑑𝑑𝑑
i.
Find
ii.
Find the gradient at (0, 15)
𝑑𝑑𝑑𝑑
(gradient func�on)
iii.
Find the gradient at (5, 10)
iv.
Find the coordinate of the point
where the gradient is 2.
2. What is the gradient of the curve with equa�on 𝑦𝑦 = 𝑥𝑥 3 − 3𝑥𝑥 2 + 4𝑥𝑥 + 7 at the point (2,-5)?
3. What is the gradient of the curve with equa�on 𝑦𝑦 = 𝑥𝑥 3 − 3𝑥𝑥 2 + 8𝑥𝑥 at the point (2,8)?
22.3 Turning points.
What are the turning points?
Turning point is a point where the gradient,
𝒅𝒅𝒅𝒅
𝒅𝒅𝒅𝒅
= 0.
There are minimum point and/or maximum point of turning point in a curve.
Ques�ons:
1. 𝑦𝑦 = 𝑥𝑥 2 − 4𝑥𝑥 + 3
a) Find
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
.
Maths
Chapter 22 Differen�a�on
b) Find the turning point of the curve.
c) State whether it is a maximum or minimum point.
2. 𝑦𝑦 = 𝑥𝑥 2 − 4𝑥𝑥 + 3
a) Find the turning point of the curve.
b) Is it a maximum or minimum point?
3. The curve 𝑦𝑦 = 𝑥𝑥 3 + 1.5𝑥𝑥 2 − 18𝑥𝑥 has two turning points. Find their x-coordinates.
4. 𝑦𝑦 = 𝑥𝑥 3 − 3𝑥𝑥 2 . Find the turning points of the curve and shown on the graph.