Let’s Review!

Using the theorems on differentiation, find the derivatives of the following

1. 𝐷𝑥 8 = 0 1
6. 𝐷𝑥 𝑥 = 𝑥2
2. 𝐷𝑥 −10 = 0 1 1−1 7. 𝐷𝑥 sin 𝑥 = cos 𝑥
= 𝑥2
2
3. 𝐷𝑥 𝑥 8 = 8𝑥 7 1 −1 8. 𝐷𝑥 tan 𝑥 = sec 2 𝑥
= 𝑥 2
2
4. 𝐷𝑥 5𝑥 = 5
1
= 1
5. 𝐷𝑥 3𝑥 3 = 9𝑥 2 2𝑥 2
Quick Review of Composite Functions
A function is composite if you can write it as f(g(x)).
 It is a function within a function, or a function of a function.

cos(𝒙𝟐 )
 because if we let f(x) = cos(x) and g(x) = 𝑥 2 , then cos(𝑥 2 ) = f(g(x)).
 g is the function within f, so we call g the “inner” function and f the
“outer” function inner

2
cos(𝑥 )
outer
Is the function a COMPOSITE or NOT COMPOSITE?
inner

1. y = (𝑥 2 +3)3 Composite

2. y = 3x + 2 Not Composite

3. y = 𝑥2 + 1 Composite
1
= (𝑥 2 +1) 2

inner
 Consider differentiating the function,

𝑦 = (2𝑥 + 2)2

What if you are given a function like

𝑦 = (2𝑥 + 2)(2𝑥 + 2)
10
𝑦 = 4𝑥 2 + 4𝑥 + 4𝑥 + 4 𝑦 = (2𝑥 + 2)
𝑦 = 4𝑥 2 + 8𝑥 + 4 Is it still convenient to take its derivative?

𝑦 ′ = 8𝑥 + 8

This method is one way of getting

the derivative of a function.
BASIC CALCULUS

CHAIN RULE AND HIGHER-ORDER

DERIVATIVE

By: Arlaine Joy L. Ke-e

 TASK GUIDE

These are your targets before moving to the

next lesson. You must focus on

 illustrating the chain rule of differentiation

 solving problems using chain rule and compute higher-
order derivatives of functions
THE CHAIN RULE
The Chain Rule tells us how to find the derivative of a composite
function or to differentiate composite functions.

If the function g(x) is differentiable at x and the function f(x) is differentiable at g(x), then the
composite function F(x) = f(g(x)) = (fog)(x) is differentiable at x, and
𝐹′ 𝑥 = 𝑓 ′ 𝑔 𝑥 • 𝑔′ 𝑥 .

If y = f(u) and u = g(x) are both differentiable functions of x, then,

𝑑𝑦 𝑑𝑦 𝑑𝑢
𝐹′ 𝑥 = = • .
𝑑𝑥 𝑑𝑢 𝑑𝑥
Remarks on Chain Rule

1. We refer to f(x) as the outer function and g(x) as the

inner function, To apply the Chain Rule, we
differentiate the outer function and multiply this to the
derivative of the inner function.

2. The Differentiation Rules can be combined with the

Chain Rule.
 Theorems in Finding the Derivatives of Composite Functions
using Chain Rule
Let u = g(x) be a differentiable function of x. Suppose the six trigonometric
functions are defined at u. Then,
𝑑𝑢 𝑑𝑢
1. 𝐷𝑥 𝑢𝑛 = 𝑛𝑢 𝑛−1
𝑑𝑥
, 𝑛 ≠ −1 5. 𝐷𝑥 cot 𝑢 = −𝑐𝑠𝑐 2 𝑢 𝑑𝑥

𝑑𝑢 𝑑𝑢
2. 𝐷𝑥 sin 𝑢 = cos 𝑢 𝑑𝑥 6. 𝐷𝑥 sec 𝑢 = sec u tan 𝑢 𝑑𝑥

𝑑𝑢 𝑑𝑢
3. 𝐷𝑥 cos 𝑢 = − sin 𝑢 7. 𝐷𝑥 cos 𝑢 = − csc 𝑢 cot 𝑢
𝑑𝑥 𝑑𝑥

2 𝑑𝑢 𝑑𝑢
4. 𝐷𝑥 tan 𝑢 = 𝑠𝑒𝑐 𝑢 8. 𝐷𝑥 𝑒𝑢 = 𝑒 𝑢
𝑑𝑥 𝑑𝑥
Decomposition of a Composite Function

𝒚 = 𝒇(𝒈 𝒙 ) u=𝒈 𝒙 𝒚 = 𝒇(𝒖)

𝑦 = (2𝑥 + 3)4 u = 2𝑥 + 3 𝑦 = 𝑢4
𝑦 = sin 2𝑥 u = 2𝑥 𝑦 = sin 𝑢

𝑦= 3𝑥 2 − 𝑥 + 1 u = 3𝑥 2 − 𝑥 + 1 𝑦= 𝑢

𝑦 = 𝑡𝑎𝑛2 𝑥 u = tan 𝑥 𝑦 = 𝑢2
= (tan 𝑥)2
 EXAMPLE:
1. Find the derivative of 𝒚 = (𝒙 + 𝟒)𝟓
Note: To apply the Chain Rule on composite functions, you must take the derivative of its outside
function and then multiply it to the derivative of its inside function.

SOLUTION:
Substitute:
𝒏−𝟏 𝒅𝒖
Apply 𝑫𝒙 𝒖 𝒏
= 𝒏𝒖 (𝟏)
𝒅𝒙
𝑑𝑦 4 𝑑𝑢
= 5𝑢
Let 𝑢 = 𝑥 + 4 𝑑𝑥 𝑑𝑥
𝑑𝑢 𝑑𝑦
=1 = 5(𝑥 + 4)4 (1)
𝑑𝑥 𝑑𝑥

y = 𝑢5 𝑑𝑦
=5 𝑥+4 4
𝑑𝑦 4 𝑑𝑢 𝑑𝑥
= 5𝑢
𝑑𝑥 𝑑𝑥
 EXAMPLE:
2. Find the derivative of 𝒚 = (𝟐𝒙 + 𝟐)𝟏𝟎
SOLUTION:

𝒅𝒖
Apply 𝑫𝒙 𝒖𝒏 = 𝒏𝒖𝒏−𝟏 (𝟏) Substitute:
𝒅𝒙
𝑑𝑦 9 𝑑𝑢
Let u = 2𝑥 + 2 = 10𝑢
𝑑𝑥 𝑑𝑥
𝑑𝑢
=2 𝑑𝑦
𝑑𝑥 = 10 2𝑥 + 2 9 (2)
𝑑𝑥
y = 𝑢10 𝑑𝑦 9
𝑑𝑦
= 20 2𝑥 + 2
9 𝑑𝑢 𝑑𝑥
= 10𝑢
𝑑𝑥 𝑑𝑥
 EXAMPLE:
3. Find the derivative of 𝒚 = (𝟑𝒙 − 𝒙𝟑 )𝟓𝟎

SOLUTION: Substitute:
𝒅𝒖
Apply 𝑫𝒙 𝒖𝒏 = 𝒏𝒖𝒏−𝟏 𝒅𝒙 (𝟏) 𝑑𝑦 49 𝑑𝑢
= 50𝑢
𝑑𝑥 𝑑𝑥
Let u = 3x - 𝑥 3 𝑑𝑦
𝑑𝑢 = 50 3x − 𝑥 3 49
(3 − 3𝑥 2 )
= 3 - 3𝑥 2 𝑑𝑥
𝑑𝑥
𝑑𝑦
= 150 − 150𝑥 2 3x − 𝑥 3 49
𝑑𝑥
y = 𝑢50
𝑑𝑦 49 𝑑𝑢
= 50𝑢
𝑑𝑥 𝑑𝑥
 SEATWORK:
Find the derivative of 𝒚 = (𝒙𝟐 + 𝟏)𝟑

SOLUTION:
𝒅𝒖
Apply 𝑫𝒙 𝒖𝒏 = 𝒏𝒖𝒏−𝟏
(𝟏)
𝒅𝒙 Substitute:
2 𝑑𝑦 2 𝑑𝑢
Let u = 𝑥 + 1 = 3𝑢
𝑑𝑢 𝑑𝑥 𝑑𝑥
= 2x 𝑑𝑦
𝑑𝑥
= 3 𝑥 2 + 1 2 (2x)
𝑑𝑥
y = 𝑢3 𝑑𝑦
𝑑𝑦 2 𝑑𝑢
= 6𝑥 𝑥 2 + 1 2
= 3𝑢 𝑑𝑥 𝑑𝑥
𝑑𝑥
 EXAMPLE:
4. Find the derivative of 𝒚 = (𝟐𝒙𝟐 + 𝟑𝒙 − 𝟓)𝟕
SOLUTION:

𝒅𝒖
Apply 𝑫𝒙 𝒖𝒏 = 𝒏𝒖𝒏−𝟏 𝒅𝒙 (𝟏)
Substitute:
Let u = 2𝑥 2 + 3𝑥 − 5 𝑑𝑦
= 7𝑢 6 𝑑𝑢
𝑑𝑢 𝑑𝑥 𝑑𝑥
= 4𝑥 + 3
𝑑𝑥 𝑑𝑦
= 7 2𝑥 2 + 3𝑥 − 5 6 (4𝑥 + 3)
𝑑𝑥
y = 𝑢7
𝑑𝑦
𝑑𝑦
= 7𝑢 6 𝑑𝑢 = 28𝑥 + 21 2𝑥 2 + 3𝑥 − 5 6
𝑑𝑥
𝑑𝑥 𝑑𝑥
 EXAMPLE:
5. Find the derivative of 𝒚 = (𝟓𝒙𝟐 + 𝟐𝒙 − 𝟏)𝟒
SOLUTION:

𝒅𝒖
Apply 𝑫𝒙 𝒖𝒏 = 𝒏𝒖𝒏−𝟏 𝒅𝒙 (𝟏)
Substitute:
Let u = 5𝑥 2 + 2𝑥 − 1 𝑑𝑦
= 4𝑢 3 𝑑𝑢
𝑑𝑢 𝑑𝑥 𝑑𝑥
= 10𝑥 + 2
𝑑𝑥 𝑑𝑦
= 4 5𝑥 2 + 2𝑥 − 1 3 (10𝑥 + 2)
𝑑𝑥
y = 𝑢4 𝑑𝑦
𝑑𝑦 = 40𝑥 + 8 5𝑥 2 + 2𝑥 − 1 3
= 4𝑢3 𝑑𝑢 𝑑𝑥
𝑑𝑥 𝑑𝑥
 SEATWORK:
Find the derivative of 𝒚 = (𝟐𝒙𝟒 − 𝟐𝒙𝟑 + 𝟑)𝟒
SOLUTION:
𝒅𝒖
Apply 𝑫𝒙 𝒖𝒏 = 𝒏𝒖𝒏−𝟏 𝒅𝒙 (𝟏)
Substitute:
Let u = 2𝑥 4 − 2𝑥 3 + 3 𝑑𝑦 3 𝑑𝑢
𝑑𝑢 = 4𝑢
= 8𝑥 3 − 6𝑥 2 𝑑𝑥 𝑑𝑥
𝑑𝑥 𝑑𝑦
= 4 2𝑥 4 − 2𝑥 3 + 3 3 (8𝑥 3 − 6𝑥 2 )
𝑑𝑥
y = 𝑢4 𝑑𝑦
𝑑𝑦 3 𝑑𝑢
= 32𝑥 3 − 24𝑥 2 2𝑥 4 − 2𝑥 3 + 3 3
= 4𝑢 𝑑𝑥
𝑑𝑥 𝑑𝑥
 EXAMPLE:
6. Find the derivative of 𝒚 = 𝒙 − 𝟑
SOLUTION:

𝒏−𝟏 𝒅𝒖
Apply 𝑫𝒙 𝒖𝒏
= 𝒏𝒖 𝒅𝒙
(𝟏) Substitute:
𝟏
𝒚= 𝒙−𝟑 𝒚 = (𝒙 − 𝟑)𝟐 𝑑𝑦 1 −1 𝑑𝑢
= 𝑢 2
𝑑𝑥 2 𝑑𝑥
1
Let u = 𝑥 − 3 𝑑𝑦 1 −
= 𝑥 − 3 2 (1)
𝑑𝑢 𝑑𝑥 2
=1 𝑑𝑦 1 −
1
𝑑𝑥 = 𝑥−3 2
1 𝑑𝑥 2
y=𝑢 2
𝑑𝑦 1
𝑑𝑦 1 −1 𝑑𝑢
= 1
𝑑𝑥
= 𝑢 2 𝑑𝑥 2(𝑥−3)2
𝑑𝑥 2
 EXAMPLE:
7.Find the derivative of 𝒚 = 𝟐𝒙 + 𝟑
SOLUTION:
𝒅𝒖
Apply 𝑫𝒙 𝒖𝒏 = 𝒏𝒖𝒏−𝟏 𝒅𝒙 (𝟏)
Substitute:
𝟏
𝒚 = 𝟐𝒙 + 𝟑 𝒚 = (𝟐𝒙 + 𝟑)𝟐 𝑑𝑦 1 −1 𝑑𝑢
= 𝑢 2
𝑑𝑥 2 𝑑𝑥
1
Let u = 2𝑥 + 3 𝑑𝑦 1 −
= 2𝑥 + 3 2 (2)
𝑑𝑢 𝑑𝑥 2
=2 𝑑𝑦 −
1
𝑑𝑥 = 1 2𝑥 + 3 2
𝑑𝑥
1
y=𝑢 1 2 𝑑𝑦 1
𝑑𝑦 1 − 𝑑𝑢 = 1
𝑑𝑥
= 𝑢 2 (2𝑥+3)2
𝑑𝑥 2 𝑑𝑥
 EXAMPLE:
8. Find the derivative of 𝒚 = 𝟓 + 𝟑𝒙𝟐
SOLUTION:
𝒏−𝟏 𝒅𝒖
Apply 𝑫𝒙 𝒖 𝒏
= 𝒏𝒖 (𝟏)
𝒅𝒙 Substitute:
𝟏
𝒚= 𝟓+ 𝟑𝒙𝟐 𝒚 = (𝟓 𝟐
+ 𝟑𝒙 )𝟐 𝑑𝑦 1 −1 𝑑𝑢
= 𝑢 2
𝑑𝑥 2 𝑑𝑥
Let u = 5 + 3𝑥 2 𝑑𝑦 1 −
1
= 5 + 3𝑥 2 2 (6𝑥)
𝑑𝑢 𝑑𝑥 2
= 6𝑥 𝑑𝑦 −
1
𝑑𝑥 = 3𝑥 5 + 3𝑥 2 2
𝑑𝑥
1
𝑑𝑦 3𝑥
y=𝑢 1 2 = 1
𝑑𝑦 1 − 𝑑𝑢 𝑑𝑥
= 𝑢 2 (5+3𝑥 2 )2
𝑑𝑥 2 𝑑𝑥
 SEATWORK:
Find the derivative of 𝒚 = 𝟔𝒙𝟐 − 𝟓
SOLUTION:
𝒅𝒖
Apply 𝑫𝒙 𝒖𝒏 = 𝒏𝒖𝒏−𝟏
𝒅𝒙
(𝟏) Substitute:
𝟏
𝒚= (𝟔𝒙𝟐 − 𝟓)𝟐 𝑑𝑦 1 −1 𝑑𝑢
𝒚= 𝟔𝒙𝟐 − 𝟑 = 𝑢 2
𝑑𝑥 2 𝑑𝑥
Let u = 6𝑥 2 − 5 𝑑𝑦 1 −
1
= 6𝑥 2 − 5 2 (12𝑥)
𝑑𝑢 𝑑𝑥 2
= 12𝑥 𝑑𝑦 −
1
𝑑𝑥 = 6𝑥 6𝑥 2 − 5 2
𝑑𝑥
1
𝑑𝑦 6𝑥
y=𝑢 1 2 = 1
𝑑𝑦 1 − 𝑑𝑢 𝑑𝑥
(6𝑥 2 −5)2
= 𝑢 2 𝑑𝑥
𝑑𝑥 2
 EXAMPLE:
9. Find the derivative of 𝒚 = 𝒔𝒊𝒏(𝟑𝒙)
SOLUTION:
𝒅𝒖
Apply 𝑫𝒙 𝒔𝒊𝒏 𝒖 = 𝒄𝒐𝒔 𝒖 𝒅𝒙 (2)
Substitute:
Let u = 3𝑥
𝑑𝑢 𝑑𝑦 𝑑𝑢
=3 = cos 𝑢
𝑑𝑥 𝑑𝑥 𝑑𝑥
𝑑𝑦
y = sin 𝑢 = cos(3𝑥)(3)
𝑑𝑥
𝑑𝑦 𝑑𝑢 𝑑𝑦
= cos 𝑢 = 3 cos(3𝑥)
𝑑𝑥 𝑑𝑥 𝑑𝑥
 EXAMPLE:
10. Find the derivative of 𝒚 = 𝒄𝒐𝒔(𝟗𝒙)
SOLUTION:

𝒅𝒖
Apply 𝑫𝒙 𝐜𝐨𝐬 𝒖 = −𝐬𝐢𝐧 𝒖 𝒅𝒙 (3)
Substitute:
Let u = 9𝑥 𝑑𝑦 𝑑𝑢
𝑑𝑢 = −sin 𝑢
=9 𝑑𝑥 𝑑𝑥
𝑑𝑥
𝑑𝑦
= −𝑠𝑖𝑛(9𝑥)(9)
y = cos 𝑢 𝑑𝑥
𝑑𝑦 𝑑𝑢 𝑑𝑦
= −sin 𝑢 = −9 sin(9𝑥)
𝑑𝑥 𝑑𝑥 𝑑𝑥
 SEATWORK:
Find the derivative of 𝒚 = 𝒕𝒂𝒏(𝟒𝒙)
SOLUTION:

𝒅𝒖
Apply 𝑫𝒙 𝐭𝐚𝐧 𝒖 = 𝒔𝒆𝒄𝟐 𝒖 𝒅𝒙 (4)
Substitute:
Let u = 4𝑥 𝑑𝑦 𝑑𝑢
2
𝑑𝑢 = 𝑠𝑒𝑐 𝑢
=4 𝑑𝑥 𝑑𝑥
𝑑𝑥
𝑑𝑦
= 𝑠𝑒𝑐 2 (4𝑥)(4)
y = tan 𝑢 𝑑𝑥
𝑑𝑦 2 𝑑𝑢 𝑑𝑦
= 𝑠𝑒𝑐 𝑢 𝑑𝑥 = 4 𝑠𝑒𝑐 2 (4𝑥)
𝑑𝑥 𝑑𝑥
 EXAMPLE: Apply 𝑫𝒙 𝒔𝒊𝒏 𝒖 = 𝒄𝒐𝒔 𝒖 𝒅𝒙 (2)
𝒅𝒖
𝒅𝒚
11. Find :𝒚 = 𝟐 𝒔𝒊𝒏(𝟑𝒙) − 𝒄𝒐𝒔𝟑 (𝟔𝒙) 𝒅𝒖
𝒅𝒙 𝑫𝒙 𝒖𝒏 = 𝒏𝒖𝒏−𝟏 (𝟏)
𝒅𝒙
SOLUTION:
first term: 2 sin(3𝑥) second term: 𝑐𝑜𝑠 3 (6𝑥) (cos 6𝑥 )3 Recall:
Let u = 3𝑥 y = 2 sin 𝑢 Let u = cos 6𝑥 y = 𝑢3 𝐷𝑥 (sin u) = cos u
𝑑𝑢 𝑑𝑦 𝒅𝒖 𝑑𝑢 𝑑𝑦 2 𝒅𝒖 𝐷𝑥 (cos u) = -sin u
=3 = 2 cos 𝑢 𝑑𝑥
= − sin 6𝑥 = 3𝑢
𝑑𝑥 𝑑𝑥 𝒅𝒙 𝑑𝑥 𝒅𝒙 𝐷𝑥 (tan u) = cos u

substitute: 𝑑𝑦 = 𝑑𝑢 substitute: 𝑑𝑦 2 𝑑𝑢
2 cos𝑢 = 3𝑢
𝑑𝑥 𝑑𝑥 𝑑𝑥 𝑑𝑥
= 2 cos(3𝑥) (3) 𝑑𝑢
= 3(cos 6𝑥)2 𝑑𝑥
= 6 cos(3𝑥)
= 3 (cos 6x)2 • [ − sin 6𝑥 6 ]
𝑑𝑦 = 3 (cos 6x)2 • −6 (sin 6𝑥)
𝑑𝑥
= [6 cos(3𝑥)] − −18 𝑐𝑜𝑠 2 (6𝑥) sin(6𝑥)
= −18 cos 6x 2 (sin 6𝑥)
𝑑𝑦
= 6 cos(3𝑥) + 18 𝑐𝑜𝑠 2 (6𝑥) sin(6𝑥) = −18 𝑐𝑜𝑠 2 (6𝑥) sin(6𝑥)
𝑑𝑥
 EXAMPLE: Apply 𝑫𝒙 𝐜𝐨𝐬 𝒖 = −𝐬𝐢𝐧 𝒖 𝒅𝒙 (3)
𝒅𝒖
𝒅𝒚
12. Find 𝒅𝒙
: 𝒚 = 𝟑 𝒄𝒐𝒔(𝟐𝒙) − 𝒔𝒊𝒏𝟒 (𝟓𝒙) 𝑫𝒙 𝒖𝒏 = 𝒏𝒖𝒏−𝟏 𝒅𝒖
(𝟏)
𝒅𝒙
SOLUTION:
3 𝑐𝑜𝑠(2𝑥)
first term: second term: 𝑠𝑖𝑛4 (5𝑥) ((sin 5𝑥)4 ) Recall:
Let u = 2𝑥 y = 3 cos 𝑢 𝐷𝑥 (sin u) = cos u
𝑑𝑦
Let u = sin 5𝑥 y = 𝑢4
𝑑𝑢
= −3 sin 𝑢 𝒅𝒖 𝑑𝑢 𝑑𝑦 𝐷𝑥 (cos u) = -sin u
=2 𝑑𝑥 𝒅𝒙 = cos 5𝑥 = 4𝑢3
𝒅𝒖
𝑑𝑥 𝑑𝑥 𝑑𝑥 𝐷𝑥 (tan u) = cos u
𝒅𝒙
substitute: 𝑑𝑦 = −3 sin 𝑢 𝑑𝑢 substitute: 𝑑𝑦 3 𝑑𝑢
𝑑𝑥 𝑑𝑥 = 4𝑢
𝑑𝑥 𝑑𝑥
= −3 sin(2𝑥)(2)
= −6 sin(2𝑥) = 4(sin 5𝑥)3 • [ cos 5𝑥 5 ]
= 4(sin 5x)3 • 5(cos 5𝑥)
𝑑𝑦
= −6 sin(2𝑥) − 20 𝑠𝑖𝑛3 (5𝑥) cos(5𝑥) = 20 𝑠𝑖𝑛3 (5𝑥) cos(5𝑥)
𝑑𝑥
= −6 sin(2𝑥) − 20 𝑠𝑖𝑛3 (5𝑥) cos(5𝑥)
 SEATWORK:
𝒅𝒖
𝒅𝒚 Apply 𝑫𝒙 𝐭𝐚𝐧 𝒖 = 𝒔𝒆𝒄𝟐 𝒖 (4)
Find 𝒅𝒙
:𝒚 = 𝟐 𝒕𝒂𝒏 𝟐𝒙 + 𝒔𝒊𝒏𝟓 (𝟑𝒙) 𝒅𝒙
𝒏−𝟏 𝒅𝒖
𝑫𝒙 𝒖𝒏 = 𝒏𝒖 (𝟏)
SOLUTION: 𝒅𝒙

2 𝑡𝑎𝑛(2𝑥)
first term: second term: 𝑠𝑖𝑛5 (3𝑥) ((sin 3𝑥)5 ) Recall:
Let u = 2𝑥 y = 2 tan 𝑢 Let u = sin 3𝑥 y = 𝑢5 𝐷𝑥 (sin u) = cos u
𝑑𝑢 𝑑𝑦 2𝑢 𝒅𝒖 𝑑𝑢 𝑑𝑦 4 𝒅𝒖 𝐷𝑥 (cos u) = -sin u
=2 = 2 𝑠𝑒𝑐 = cos 3𝑥 = 5𝑢
𝑑𝑥 𝑑𝑥 𝒅𝒙 𝑑𝑥 𝑑𝑥 𝒅𝒙 𝐷𝑥 (tan u) = 𝑠𝑒𝑐 2 u

substitute: 𝑑𝑦 𝑑𝑢 substitute: 𝑑𝑦 𝑑𝑢
= 2 𝑠𝑒𝑐 2 𝑢 = 5𝑢 4
𝑑𝑥 𝑑𝑥 𝑑𝑥 𝑑𝑥
= 2 𝑠𝑒𝑐 2 (2𝑥) (2) = 5(sin 3𝑥)4 • [ cos 3𝑥 3 ]
= 4 𝑠𝑒𝑐 2 (2𝑥) = 5(sin 3x)4 • 3(cos 3𝑥)
𝑑𝑦 = 15 𝑠𝑖𝑛4 (3𝑥) (cos 3𝑥)
= 4 𝑠𝑒𝑐 2 2𝑥 + 15 𝑠𝑖𝑛4 (3𝑥) (cos 3𝑥)
𝑑𝑥
= 4 𝑠𝑒𝑐 2 2𝑥 + 15 𝑠𝑖𝑛4 (3𝑥) (cos 3𝑥)
 EXAMPLE:
𝒙𝟐
13. Find the derivative of 𝒚 = 𝟕𝒆
SOLUTION:

𝒅𝒖
Apply 𝑫𝒙 𝒆𝒖 = 𝒆𝒖 𝒅𝒙 (8)
Substitute:
Let u = 𝑥 2 𝑑𝑦 𝑢 𝑑𝑢
𝑑𝑢
= 7𝑒
𝑑𝑥 𝑑𝑥
= 2x
𝑑𝑥
𝑑𝑦 𝑥2
𝑢
= 7𝑒 (2𝑥)
y = 7𝑒 𝑑𝑥
𝑑𝑦 𝑢 𝑑𝑢
= 7𝑒
𝑑𝑥 𝑑𝑥
 EXAMPLE:
𝟔𝒙𝟐
14. Find the derivative of 𝒚 = 𝟐𝒆
SOLUTION:
𝒅𝒖
Apply 𝑫𝒙 𝒆𝒖 = 𝒖
𝒆 𝒅𝒙 (8)
Substitute:
2
Let u = 6𝑥 𝑑𝑦 𝑢 𝑑𝑢
𝑑𝑢 = 2𝑒
= 12x 𝑑𝑥 𝑑𝑥
𝑑𝑥
𝑑𝑦 6𝑥 2
= 2𝑒 (12𝑥)
y = 2𝑒 𝑢 . 𝑑𝑥
𝑑𝑦
= 2𝑒 𝑢 𝑑𝑢
𝑑𝑥 𝑑𝑥
 EXAMPLE:
𝒙𝟐 −𝟒𝒙
15. Find the derivative of 𝒚 = 𝟓𝒆
SOLUTION:

𝒖 𝒅𝒖
Apply 𝑫𝒙 𝒆 𝒖
= 𝒆 𝒅𝒙 (8) Substitute:
𝑑𝑦 𝑢 𝑑𝑢
= 5𝑒
Let u = 𝑥 2 − 4𝑥 𝑑𝑥 𝑑𝑥
𝑑𝑢
= 2x - 4 𝑑𝑦
= 5𝑒 𝑥 2 −4𝑥
(2𝑥 − 4)
𝑑𝑥
𝑑𝑥

y = 5𝑒 𝑢
𝑑𝑦
= 5𝑒 𝑢 𝑑𝑢
𝑑𝑥 𝑑𝑥
 EXAMPLE:
𝒙𝟓 +𝟏𝟎𝒙
16. Find the derivative of 𝒚 = 𝟐𝒆
SOLUTION:

𝒅𝒖
Apply 𝑫𝒙 𝒆𝒖 = 𝒆𝒖 𝒅𝒙 (8)
Substitute:
Let u = 𝑥 5 + 10𝑥 𝑑𝑦 𝑢 𝑑𝑢
= 2𝑒
𝑑𝑢 𝑑𝑥 𝑑𝑥
= 5𝑥 4 + 10
𝑑𝑥
𝑑𝑦 𝑥 5 +10𝑥
= 2𝑒 (5𝑥 4 + 10)
y = 2𝑒 𝑢 𝑑𝑥
𝑑𝑦 𝑢 𝑑𝑢
= 2𝑒
𝑑𝑥 𝑑𝑥
 SEATWORK:
𝟑𝒙𝟑 +𝟐𝒙
Find the derivative of 𝒚 = 𝟏𝟎𝒆
SOLUTION:

𝒅𝒖
Apply 𝑫𝒙 𝒆𝒖 = 𝒆𝒖 𝒅𝒙 (8)
Substitute:
Let u = 3𝑥 3 + 2𝑥 𝑑𝑦 𝑢 𝑑𝑢
𝑑𝑢
= 9𝑥 2 + 2 = 10𝑒
𝑑𝑥
𝑑𝑥 𝑑𝑥

𝑑𝑦 3𝑥 3 +2𝑥
y = 10𝑒 .𝑢
𝑑𝑥
= 10𝑒 (9𝑥 2 + 2)
𝑑𝑦
= 10𝑒 𝑢 𝑑𝑢
𝑑𝑥 𝑑𝑥
 EXAMPLE: 𝟒𝒙
17. Find the derivative of 𝐲 =
1 𝟑𝒙
4𝑥 4𝑥 2 𝒅𝒖
SOLUTION: 𝑦= 𝑦= Apply 𝑫𝒙 𝒖𝒏 = 𝒏𝒖𝒏−𝟏
(𝟏)
3𝑥 3𝑥 𝒅𝒙

Let u =
4𝑥 Substitute:
3𝑥 𝑑𝑦 1 −1 𝑑𝑢
= 𝑢 2
𝑑𝑥 2 𝑑𝑥
1
𝑑𝑦 1 4𝑥 −2 4 (3𝑥)−(4𝑥)(3)
𝑑𝑢
=
𝐷𝑥 4𝑥 •(3𝑥)−(4𝑥)•𝐷𝑥 (3𝑥) =
𝑑𝑥 2 3𝑥 (3𝑥)2
𝑑𝑥 (3𝑥)2 1
𝑑𝑢 4 (3𝑥)−(4𝑥)(3) 𝑑𝑦 1 3𝑥 2 12𝑥−12𝑥
= =
𝑑𝑥 (3𝑥)2 𝑑𝑥 2 4𝑥 (3𝑥)2
1
1 𝑑𝑦 1 3𝑥 2
y=𝑢 1 2 = (0)
𝑑𝑥 2 4𝑥
𝑑𝑦 1 − 𝒅𝒖
= 𝑢 2 𝑑𝑦
𝑑𝑥 2 𝒅𝒙 =0
𝑑𝑥
 EXAMPLE:
𝟕𝒙𝟐
18. Find the derivative of 𝐲 = 𝟑𝒙
1
SOLUTION: 𝟕𝒙𝟐 𝟕𝒙𝟐 2 Apply 𝑫𝒙 𝒖𝒏 = 𝒏𝒖𝒏−𝟏 𝒅𝒙 (𝟏)
𝒅𝒖
𝑦= 𝑦=
𝟑𝒙 𝟑𝒙
7𝑥 2 Substitute:
Let u = 3𝑥
𝑑𝑦 1 −1 𝑑𝑢
= 𝑢 2
𝑑𝑥 2 𝑑𝑥
1
𝐷𝑥 7𝑥 2 •(3𝑥)−(7𝑥 2 )•𝐷𝑥 (3𝑥) −
𝑑𝑢 𝑑𝑦 1 7𝑥 2 2 14𝑥 (3𝑥)−(7𝑥 2 )(3)
= =
𝑑𝑥 (3𝑥)2 𝑑𝑥 2 3𝑥 (3𝑥)2
1
𝑑𝑢 14𝑥 (3𝑥)−(7𝑥 2 )(3) 𝑑𝑦 1 3𝑥 42𝑥 2 −21𝑥 2
= =
2
𝑑𝑥 (3𝑥)2 𝑑𝑥 2 7𝑥 2 (3𝑥)2
1
y=𝑢 2
𝑑𝑦 1 3𝑥
1
21𝑥 2
2
𝑑𝑦 1 −1 𝒅𝒖 =
= 𝑢 2
𝒅𝒙
𝑑𝑥 2 7𝑥 2 3𝑥 2
𝑑𝑥 2
 SEATWORK: 𝟐𝒙+𝟑
Find the derivative of 𝐲 =
1
𝟓𝒙−𝟏
2𝑥 + 3 2𝑥 + 3 2
SOLUTION: 𝑦= 𝑦=
5𝑥 − 1 5𝑥 − 1
𝒅𝒖
2𝑥−3 Apply 𝑫𝒙 𝒖𝒏 = 𝒏𝒖𝒏−𝟏 (𝟏)
Let u = 𝒅𝒙
5𝑥−1
1
𝑑𝑦 1 2𝑥−3 −2 2 5𝑥−1 − 2𝑥−3 5
=
𝑑𝑥 2 5𝑥−1 1 5𝑥−1 2
𝑑𝑢 𝐷𝑥 2𝑥−3 •(5𝑥−1)−(2𝑥−3)•𝐷𝑥 (5𝑥−1) 𝑑𝑦 1 5𝑥−1 2 2 5𝑥−1 − 2𝑥−3 5
= =
𝑑𝑥 (5𝑥−1)2 𝑑𝑥 2 2𝑥−3 5𝑥−1 2
1
𝑑𝑢 2 (5𝑥−1)−(2𝑥−3)(5) 𝑑𝑦 1 5𝑥−1 10𝑥−2 −10𝑥−15
= =
2
𝑑𝑥 (5𝑥−1)2 𝑑𝑥 2 2𝑥−3 5𝑥−1 2
1
1
y=𝑢 1 2
𝑑𝑦 1 5𝑥−1 2 −17
𝑑𝑦 1 − 𝒅𝒖 =
=
𝑑𝑥 2
𝑢 2
𝒅𝒙 𝑑𝑥 2 2𝑥−3 5𝑥−1 2
Higher-order
derivatives
HIGHER-ORDER DERIVATIVES
We know that the derivative of a function is also a function. Therefore, we can again get the
derivative of a derivative. The result is also a function so we can differentiate this again. We
end up with what we call higher-order derivatives.

The previous differentiation rules apply to these derivatives. We list the notations for the
higher-order derivatives.

Given y = f(x). Then, 1st derivative: y ′ , 𝑓 ′ 𝑥 ,

𝑑𝑦
, 𝐷𝑥 𝑦, 𝐷𝑥 𝑓 𝑥 .
𝑑𝑥
𝑑2 𝑦
2nd derivative: y , 𝑓
′′ ′′
𝑥 , , 𝐷𝑥 𝑦′, 𝐷𝑥 𝑓′ 𝑥 .
𝑑𝑥 2
𝑑3 𝑦
3rd derivative: y′ , 𝑓′
′′ ′′
𝑥 , 𝑑𝑥 3
, 𝐷𝑥 𝑦′′, 𝐷𝑥 𝑓′′ 𝑥 .
∙
∙
∙
𝑑𝑛 𝑦
nth derivative: 𝑦 (𝑛)
,𝑓 (n)
𝑥 , 𝑛 , 𝐷𝑥 𝑦 𝑛−1 , 𝐷𝑥 𝑓 𝑛−1 𝑥 .
𝑑𝑥
 EXAMPLE:
1. f(x) = 2𝑥 3 , find 𝑓′′′(𝑥)

SOLUTION:

f(x) = 2𝑥 3

𝑓 ′ (x) = 6𝑥 2
𝑓 ′′ (x) = 12𝑥
𝑓 ′′′ (x) = 12
 EXAMPLE:
2. f(x) = 4𝑥 2 + 5, find 𝑓′′(𝑥)

SOLUTION:

f(x) = 4𝑥 2 + 5

𝑓 ′ (x) = 8𝑥
𝑓 ′′ (x) = 8
 EXAMPLE:
3. f(x) = 𝑥 − 6𝑥 + 17𝑥 + 8, find 𝑓′′(𝑥)
3 2

SOLUTION:

f(x) = 𝑥 3 − 6𝑥 2 + 17𝑥 + 8
𝑓 ′ (x) = 3𝑥 2 − 12𝑥 + 17
𝑓 ′′ (x) = 6𝑥 − 12
 SEATWORK:

f(x) = 4𝑥 5 + 3𝑥 4 + 𝑥 3 − 6𝑥 2 − 17𝑥 + 8, find 𝑓 (4) (𝑥)

SOLUTION:

f(x) = 4𝑥 5 + 3𝑥 4 + 𝑥 3 − 6𝑥 2 − 17𝑥 + 8
𝑓 ′ (x) = 20𝑥 4 + 12𝑥 3 + 3𝑥 2 − 12𝑥 − 17
𝑓 ′′ (x) = 80𝑥 3 + 36𝑥 2 + 6𝑥 − 12
𝑓 ′′′ (x) = 240𝑥 2 + 72𝑥 + 6
𝑓 (4) (x) = 480𝑥 + 72
 EXAMPLE:
4. y = 𝑒 4𝑥 , find 𝑦′′′

SOLUTION: 𝑦 = 𝑒 4𝑥 Recall:
𝐷𝑥 𝑒 𝑢 = 𝑒 𝑢 𝑑𝑢
𝑦′ = 𝑒 4𝑥 (4) 𝐷𝑥 cos 𝑢 = sin 𝑢 𝑑𝑢
= 4𝑒 4𝑥
𝐷𝑥 sin 𝑢 = cos 𝑢 𝑑𝑢
𝑦′′ = 4𝑒 4𝑥 (4)
= 16𝑒 4𝑥
𝑦′′′ = 16𝑒 4𝑥 (4)
= 64𝑒 4𝑥
 EXAMPLE:
5. y = 𝑒 2𝑥 + 2𝑥 3 , find 𝑦′′

SOLUTION: 2𝑥 3 Recall:
𝑦=𝑒 + 2𝑥
𝐷𝑥 𝑒 𝑢 = 𝑒 𝑢 𝑑𝑢
2𝑥 2
𝑦′ = 𝑒 (2) + 6𝑥
2𝑥 2
= 2𝑒 + 6𝑥

𝑦′′ = 2𝑒 2𝑥 (2) + 12𝑥

2𝑥
= 4𝑒 + 12𝑥
 SEATWORK:

y = 4𝑥 2 − 2𝑒 5𝑥 , find 𝑦′′′

SOLUTION: 𝑦 = 4𝑥 − 2𝑒
2 5𝑥 Recall:
𝐷𝑥 𝑒 𝑢 = 𝑒 𝑢 𝑑𝑢
𝑦′ = 8𝑥 − 2𝑒 5𝑥 (5)
= 8𝑥 − 10𝑒 5𝑥
𝑦′′ = 8 − 10𝑒 5𝑥 (5)
= 8 − 50𝑒 5𝑥

𝑦′′′ = 50𝑒 5𝑥 (5)

= 250𝑒 5𝑥
 EXAMPLE:
1
6. 𝑓 𝑥 = 2𝑥 + 6𝑥 , find 𝑓′′(𝑥)
2

1
SOLUTION:
𝑓 𝑥 = 2𝑥 + 6𝑥2
Recall:
1
1 −
𝑓′ 𝑥 = 2𝑥 2 +6
2 1
−
=𝑥 2 +6
1 −3
𝑓′′ 𝑥 = −2𝑥 2
 EXAMPLE:
7. 𝑓 𝑥 = 3 sin(5𝑥) , find 𝑓′′(𝑥)

SOLUTION: Recall:
𝑓 𝑥 = 3 sin(5x)
𝐷𝑥 sin 𝑢 = cos 𝑢 𝑑𝑢
𝑓′ 𝑥 = 3 cos(5x)(5)
= 15 cos(5x)

𝑓′′ 𝑥 = 15 –sin(5x)(5)
= –75 sin(5x)
 SEATWORK:

𝑓 𝑥 = 2 cos(4𝑥) , find 𝑓′′′(𝑥)

SOLUTION: 𝑓 𝑥 = 2 cos(4x) Recall:

𝐷𝑥 cos 𝑢 = sin 𝑢 𝑑𝑢
𝑓′ 𝑥 = 2 –sin(4x)(4)
= −8 sin(4𝑥)
𝑓′′′ 𝑥 = −32 − sin(4𝑥)(4)
𝑓′′ 𝑥 = −8 cos(4𝑥)(4)
= 128 sin(4𝑥)
= −32 cos(4𝑥)
 EXAMPLE:
𝑑2𝑦
8. y = 𝑒 2𝑥 + 3 cos 5x, find
𝑑𝑥 2
SOLUTION:
𝑑𝑦
= 𝑒 2𝑥 2 + 3(−sin 5𝑥)(5) Recall:
𝑑𝑥
𝑑𝑦 𝐷𝑥 𝑒 𝑢 = 𝑒 𝑢 𝑑𝑢
= 2𝑒 2𝑥 −3(sin 5𝑥)(5)
𝑑𝑥 𝐷𝑥 cos 𝑢 = sin 𝑢 𝑑𝑢
= 2𝑒 2𝑥 −15(sin 5𝑥)

𝑑2 𝑦
= 2𝑒 2𝑥 (2) − 15(cos 5𝑥)(5)
𝑑𝑥 2
= 4𝑒 2𝑥 −75(cos 5𝑥)
Good Job, Learners!