MATH-163
PRE MATH-II
Atifa Kanwal
atifa.kanwal@seecs.edu.pk
Office # 303, Faculty Block, SEECS, NUST
THEOREMS ON
DIFFERENTIATION
Section 2.2,2.3
𝑑 𝑛
𝑥 = 𝑛𝑥 𝑛−1
𝑑𝑥
Examples:
Find derivative of the
following functions.
5
1
100
(i) 𝑥
(ii) 6
(iii) 𝑥 3
𝑥
Solution:
𝑑
(i)
𝑥 100 = 100𝑥 99
(ii)
(iii)
𝑑𝑥
𝑑
1
𝑑𝑥 𝑥 6
5
𝑑
𝑥3
𝑑𝑥
𝑑
𝑐 =0
𝑑𝑥
𝑑
𝑥 −6 = −6𝑥 −7
𝑑𝑥
5 5−1
5 2
= 𝑥3 = 𝑥3
3
3
=
𝑑
𝑑
𝑐𝑓 𝑥 = 𝑐
𝑓 𝑥
𝑑𝑥
𝑑𝑥
Examples:
Find derivative of 2𝑥 3
Solution:
𝑑
𝑑
2𝑥 3 = 2
𝑥 3 = 6𝑥 2
𝑑𝑥
𝑑𝑥
𝑑
𝑎𝑥 + 𝑏 𝑛 = 𝑛𝑎 𝑎𝑥 + 𝑏 𝑛−1
𝑑𝑥
𝑑
1
− 𝑛−1
=
𝑛𝑎
𝑎𝑥
+
𝑏
𝑑𝑥 𝑎𝑥 + 𝑏 𝑛
Examples:
Find derivative of the
following functions.
(i) (2𝑥 + 3)100 (ii)
1
(3𝑥+1)6
Solution:
𝑑
(i)
(2𝑥 + 3)100 = 2 ×
𝑑𝑥
100(2𝑥 + 3)99 =
200(2𝑥 + 3)99
𝑑
1
(ii)
6 =
𝑑𝑥 (−3𝑥+1)
𝑑
(−3𝑥 + 1)−6
𝑑𝑥
− 6 −3𝑥 + 1 −7
= −3 ×
= 18(−3𝑥 + 1)−7
EXERCISE 2.2
𝑑
𝑓 𝑥 ±𝑔 𝑥
𝑑𝑥
𝑑
=
𝑓 𝑥
𝑑𝑥
𝑑
±
𝑔 𝑥
𝑑𝑥
Example:
3
2
Find derivative of
𝑦 = 𝑥 4 + 𝑥 3 − 2𝑥 + 5
4
3
Solution:
𝑑𝑦 3
2
3
= × 4𝑥 + × 3𝑥 2 − 2 = 3𝑥 3 + 2𝑥 2 − 2
𝑑𝑥 4
3
𝑑
𝑓 𝑥 .𝑔 𝑥
𝑑𝑥
𝑑
=𝑔 𝑥
𝑓 𝑥
𝑑𝑥
𝑑
+𝑓 𝑥
𝑔 𝑥
𝑑𝑥
Example:
Find derivative of 𝑦 = 2 𝑥 + 2 𝑥 − 𝑥
Solution:
𝑑𝑦
𝑑
𝑑
= 2 𝑥+2
𝑥− 𝑥 + 𝑥− 𝑥
2 𝑥+2
𝑑𝑥
𝑑𝑥
𝑑𝑥
1
2
= 2 𝑥+2 1−
+ 𝑥− 𝑥
+0
2 𝑥
2 𝑥
1
=3 𝑥−
𝑥
Alternatively, previous example can be solved as:
Example:
Find derivative of 𝑦 = 2 𝑥 + 2 𝑥 − 𝑥
Solution:
2
𝑥3
3
2𝑥 2
𝑦=
−2 𝑥 =2
− 𝑥
3
𝑑𝑦
𝑑
3 1
1
2
2
=2
𝑥 − 𝑥 =2 𝑥 −
𝑑𝑥
𝑑𝑥
2
2 𝑥
𝑑𝑦
1
= 3 𝑥−
𝑑𝑥
𝑥
Exercise
Find Derivative of 𝑦 w.r.t 𝑥.
1
1. 𝑦 = 𝑥 2 + 1 𝑥 + 5 +
3
4
2. 𝑦 = 1 + 2𝑥 2 𝑥 −
𝑥
𝑥 −3
𝑑 𝑓 𝑥
𝑑𝑥 𝑔 𝑥
=
𝑔 𝑥
𝑑
𝑓 𝑥
𝑑𝑥
−𝑓 𝑥
𝑔 𝑥
𝑑
𝑔 𝑥
𝑑𝑥
2
Example:
Find derivative of 𝑦 =
Solution:
𝑑𝑦
=
𝑑𝑥
𝑥 2 +1
𝑥 2 −3
𝑑 2
𝑑 2
𝑥 + 1 − 𝑥2 + 1
𝑥 −3
𝑑𝑥
𝑑𝑥
𝑥2 − 3 2
𝑥 2 − 3 2𝑥 − 𝑥 2 + 1 2𝑥
=
𝑥2 − 3 2
−8𝑥
= 2
𝑥 −3 2
𝑥2 − 3
EXERCISE 2.3