Chapter 3 Lesson 5
Chain Rule
𝑑𝑦 𝑑𝑢
𝑦′ =
∙
𝑑𝑢 𝑑𝑥
Easy Math
Mr. Ali Abdalla
2
Find the derivative of each function
2
1) 𝑦 = 𝑥 + 3
2
‫أوجد مشتقة كل دالة‬
2
2) 𝑦 = 𝑥 + 3
3
______________________________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________________________
Easy Math
Mr. Ali Abdalla
3
2
2
2
3
1) 𝑦 = 𝑥 + 3
2) 𝑦 = 𝑥 + 3
𝟐
𝟏
𝟐
𝟐
𝒚′ = 𝟐 𝒙 + 𝟑
𝒚′ = 𝟑 𝒙 + 𝟑
What do you think about derivative of
∙ 𝟐𝒙
∙ 𝟐𝒙
‫ماذا عن مشتقة كل من‬
____________________________________________________________________________________________________________________________________
3) 𝑦 = 𝑥 2 + 3
4
____________________________________________________________________________________________________________________________________
____________________________________________________________________________________________________________________________________
____________________________________________________________________________________________________________________________________
4) 𝑦 = 𝑥 2 + 3
𝑛
____________________________________________________________________________________________________________________________________
____________________________________________________________________________________________________________________________________
____________________________________________________________________________________________________________________________________
Easy Math
Mr. Ali Abdalla
4
Easy Math
Mr. Ali Abdalla
5
It is often helpful to think of the chain rule in Leibniz notation.
If 𝑦 = 𝑓(𝑢) and 𝑢 = g(𝑥), then 𝑦 = 𝑓(g(𝑥)) and the chain
rule says that
𝑑𝑦 𝑑𝑦 𝑑𝑢
=
∙
𝑑𝑥 𝑑𝑢 𝑑𝑥
Easy Math
Mr. Ali Abdalla
6
Note
For the composition 𝑓(g(𝑥)),
𝑓 is often referred to as the outside function and g is referred to as
the inside function. The chain rule derivative 𝑓′(g(𝑥))g′(𝑥) can then
be viewed as the derivative of the outside function times the
derivative of the inside function.
𝑑
𝑓 g 𝑥
= 𝑓 ′ g 𝑥 g ′ (𝑥)
𝑑𝑥
𝑑
𝑓 g 𝑥
𝑑𝑥
𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑜𝑓
𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑜𝑓
𝑰𝑵𝑺𝑰𝑫𝑬
= 𝑶𝑼𝑻𝑺𝑰𝑫𝑬 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛, ∙
𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛
𝑙𝑒𝑎𝑣𝑒 𝑖𝑛𝑠𝑖𝑑𝑒 𝑎𝑙𝑜𝑛𝑒
‫مشتقة الدالة اخلارجية‬
‫مع إبقاء الداخلية كما هي‬
Easy Math
‫مشتقة الدالة‬
‫الداخلية‬
Mr. Ali Abdalla
‫‪7‬‬
‫أوجد مشتقة‬
‫‪Remember that: chain rule says‬‬
‫𝑓𝑜 𝑒𝑣𝑖𝑡𝑎𝑣𝑖𝑟𝑒𝑑‬
‫𝑓𝑜 𝑒𝑣𝑖𝑡𝑎𝑣𝑖𝑟𝑒𝑑‬
‫𝑦𝑑‬
‫𝑬𝑫𝑰𝑺𝑵𝑰‬
‫∙ ‪= 𝑶𝑼𝑻𝑺𝑰𝑫𝑬 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛,‬‬
‫𝑥𝑑‬
‫𝑛𝑜𝑖𝑡𝑐𝑛𝑢𝑓‬
‫𝑒𝑛𝑜𝑙𝑎 𝑒𝑑𝑖𝑠𝑛𝑖 𝑒𝑣𝑎𝑒𝑙‬
‫‪Find the derivative of‬‬
‫‪3‬‬
‫‪1) 𝑦 = 𝑥 2 + 3𝑥 − 2‬‬
‫____________________________________________________________________________________________________________________________________‬
‫مشتقة الدالة‬
‫مشتقة الدالة‬
‫____________________________________________________________________________________________________________________________________‬
‫اخلارجية مع إبقاء‬
‫الداخلية‬
‫الداخلية كما هي‬
‫____________________________________________________________________________________________________________________________________‬
‫____________________________________________________________________________________________________________________________________‬
‫____________________________________________________________________________________________________________________________________‬
‫____________________________________________________________________________________________________________________________________‬
‫____________________________________________________________________________________________________________________________________‬
‫‪Mr. Ali Abdalla‬‬
‫‪Easy Math‬‬
‫أوجد‬
‫‪Remember that: chain rule says‬‬
‫𝑓𝑜 𝑒𝑣𝑖𝑡𝑎𝑣𝑖𝑟𝑒𝑑‬
‫𝑓𝑜 𝑒𝑣𝑖𝑡𝑎𝑣𝑖𝑟𝑒𝑑‬
‫𝑦𝑑‬
‫𝑬𝑫𝑰𝑺𝑵𝑰‬
‫∙ ‪= 𝑶𝑼𝑻𝑺𝑰𝑫𝑬 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛,‬‬
‫𝑥𝑑‬
‫𝑛𝑜𝑖𝑡𝑐𝑛𝑢𝑓‬
‫𝑒𝑛𝑜𝑙𝑎 𝑒𝑑𝑖𝑠𝑛𝑖 𝑒𝑣𝑎𝑒𝑙‬
‫مشتقة الدالة‬
‫الداخلية‬
‫‪2) Find‬‬
‫____________________________________________________________________‬
‫____________________________________________________________________‬
‫‪The derivative of square root function‬‬
‫مشتقة دالة اجلذر التربيعي‬
‫مشتقة ما حتت اجلذر‬
‫=‬
‫‪ × 2‬اجلذر نفسه 𝑥 𝑓 ‪2‬‬
‫𝑥‪20 + 6‬‬
‫𝑑‬
‫𝑥𝑑‬
‫____________________________________________________________________‬
‫مشتقة الدالة اخلارجية‬
‫مع إبقاء الداخلية كما هي‬
‫𝑥 ‪𝑓′‬‬
‫‪8‬‬
‫____________________________________________________________________‬
‫____________________________________________________________________‬
‫𝑥 𝑓‬
‫𝑑‬
‫𝑥𝑑‬
‫____________________________________________________________________‬
‫____________________________________________________________________‬
‫____________________________________________________________________‬
‫‪Mr. Ali Abdalla‬‬
‫‪Easy Math‬‬
‫أوجد مشتقة‬
‫‪Remember that: chain rule says‬‬
‫𝑓𝑜 𝑒𝑣𝑖𝑡𝑎𝑣𝑖𝑟𝑒𝑑‬
‫𝑓𝑜 𝑒𝑣𝑖𝑡𝑎𝑣𝑖𝑟𝑒𝑑‬
‫𝑦𝑑‬
‫𝑬𝑫𝑰𝑺𝑵𝑰‬
‫∙ ‪= 𝑶𝑼𝑻𝑺𝑰𝑫𝑬 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛,‬‬
‫𝑥𝑑‬
‫𝑛𝑜𝑖𝑡𝑐𝑛𝑢𝑓‬
‫𝑒𝑛𝑜𝑙𝑎 𝑒𝑑𝑖𝑠𝑛𝑖 𝑒𝑣𝑎𝑒𝑙‬
‫مشتقة الدالة‬
‫الداخلية‬
‫مشتقة الدالة اخلارجية‬
‫مع إبقاء الداخلية كما هي‬
‫‪3/2‬‬
‫‪9‬‬
‫‪3) Find the derivative of:‬‬
‫‪3‬‬
‫‪𝑓 𝑥 = 2𝑥 + 5‬‬
‫____________________________________________________________________‬
‫____________________________________________________________________‬
‫____________________________________________________________________‬
‫____________________________________________________________________‬
‫____________________________________________________________________‬
‫____________________________________________________________________‬
‫____________________________________________________________________‬
‫____________________________________________________________________‬
‫‪Mr. Ali Abdalla‬‬
‫‪Easy Math‬‬
10
3) Find the derivative of:
𝑓 𝑥 =
2𝑥 + 5
3𝑥 − 1
5
‫أوجد مشتقة‬
Remember that: chain rule says
𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑜𝑓
𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑜𝑓
𝑑𝑦
𝑰𝑵𝑺𝑰𝑫𝑬
= 𝑶𝑼𝑻𝑺𝑰𝑫𝑬 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛, ∙
𝑑𝑥
𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛
𝑙𝑒𝑎𝑣𝑒 𝑖𝑛𝑠𝑖𝑑𝑒 𝑎𝑙𝑜𝑛𝑒
____________________________________________________________________
‫مشتقة الدالة اخلارجية‬
‫مع إبقاء الداخلية كما هي‬
____________________________________________________________________
____________________________________________________________________
‫مشتقة الدالة‬
‫الداخلية‬
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
Remember that: Quotient rule says
____________________________________________________________________
____________________________________________________________________
Easy Math
𝑑 𝑓 𝑥
𝑑𝑥 g 𝑥
𝑓 ′ 𝑥 g 𝑥 − 𝑓 𝑥 g′ 𝑥
=
g 𝑥 2
Mr. Ali Abdalla
11
Theorem 5.2
If 𝑓 is differentiable everywhere on its domain and has an inverse
function g = 𝑓 −1 then
1
g′ 𝑥 = ′
𝑓 g 𝑥
for all 𝑥 in the domain of g, provided 𝑓 ′ g 𝑥
Easy Math
≠0
Mr. Ali Abdalla
12
1
Given that the function 𝑓 𝑥 = 𝑥 5 + 3𝑥 3 + 2𝑥 + 1 has an inverse
function g 𝑥 , compute g ′ 7 ?
Remember that:
Solution
Step 1:
g′
Where 𝑓(𝑥) and g(𝑥) are inverses
and g 7 = 𝑥 then
𝑥 =
g′
7 =
1
𝑓′ g 7
Remember that:
3
𝑥 + 3𝑥 + 2𝑥 + 1 = 7
Therefore
𝑓′ g 𝑥
Remember that:
𝑓 𝑥 =7
5
1
g′
𝑥=1
g 7 =𝑥=1
1
7 = ′
𝑓 1
Continue
Easy Math
Mr. Ali Abdalla
13
Step 2:
Find the derivative of 𝑓(𝑥)
Remember that:
𝑓 ′ 𝑥 = 5𝑥 4 + 9𝑥 2 + 2
𝑓′ 1 = 5 1
= 16
Step 3:
′
4
+9 1
2
+2
g′ 7 =
1
𝑓′ g 7
Remember that:
g′ 7 =
1
𝑓′ 1
Use the formula
g 7 =
1
𝑓′ g 7
1
1
= ′
=
𝑓 1
16
Easy Math
Mr. Ali Abdalla
14
g 2 ‫ احسب‬. g 𝑥 , ‫ 𝑥 = 𝑥 𝑓 هلا دالة عكسية‬+ 7𝑥 + 2 ‫على فرض أن الدالة‬
′
2
3
Given that the function 𝑓 𝑥 = 𝑥 3 + 7𝑥 + 2 has an inverse function
g 𝑥 , compute g ′ 2 ?
______________________________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________________________
Easy Math
Mr. Ali Abdalla
15
‫ و‬g 1 = 2 ‫ 𝑓 و‬1 = 3 ‫ حيث‬ℎ′(1) ‫ احسب‬ℎ(𝑥) = 𝑓(g(𝑥)) ‫إذا كان‬
g′(3) = 5 ‫ و‬g′(1) = −2 ‫ 𝑓 و‬′ 2 = 3 ‫ 𝑓 و‬′ 1 = 4
3
3
If ℎ(𝑥) = 𝑓(g(𝑥)) compute the ℎ′(1), where 𝑓 1 = 3, g 1 = 2
, 𝑓 ′ 1 = 4, 𝑓 ′ 2 = 3, g′(1) = −2 and g′(3) = 5
____________________________________________________________________________________________________________________________________
____________________________________________________________________________________________________________________________________
____________________________________________________________________________________________________________________________________
____________________________________________________________________________________________________________________________________
____________________________________________________________________________________________________________________________________
____________________________________________________________________________________________________________________________________
____________________________________________________________________________________________________________________________________
Easy Math
Mr. Ali Abdalla
‫شكراً لمك‬
‫اىل اللقاء يف الفيديو القادم‬
‫ومتارين منوعة‬
‫‪Mr. Ali Abdalla‬‬
‫‪Easy Math‬‬