Rules of Derivatives to solve the economic equations

1. Derivatives
Rules of Derivatives

2. Derivative
• A small change in dependent variable in
response to a small change in its independent
variable.
• Y= F(x)
• Y = dependent variable
• X= independent variable
• For example:
• Y = 2𝑥2
+2𝑥 + 5
• Y = F(1) = 2(1)+2(1)+5 = 9
• Y =F(2)= 2(2)2
+ 2(2)+5 = 8+4+5= 17

3. Rules of Derivative
• Constant rule
• Variable rule
• Power rule
• Sum and difference rule
• Derivative of a derivative
• Product rule
• Quotient rule

4. • Constant rule
Y= 100
𝑑𝑦
𝑑𝑥
or f’(x)= 0
• Variable rule
Y = x ;
𝑑𝑦
𝑑𝑥
= 1𝑥1−1
= 1𝑋0
=1(1)
Y= 1

5. Power rule
If there is a power in the function the power is
multiplied by coefficient.
Y= 4𝑥6
Y=
𝑑𝑦
𝑑𝑥
= 6(4)𝑥6−1
Y=
𝑑𝑦
𝑑𝑥
= 24𝑥5
Y= 24𝑥5

6. Sum/ Difference rule
• Y = 5𝑥4
+40
𝑑𝑦
𝑑𝑥
= 𝑑(5𝑥4
) + 𝑑 40
𝑑𝑦
𝑑𝑥
= 20𝑥4−1
+ 0
𝑑𝑦
𝑑𝑥
= 20𝑥3

7. Derivative of a higher derivative
•
𝑑𝑦
𝑑𝑥
= 20𝑥3
•
𝑑𝑦2
𝑑𝑥2 = 60𝑥3−1
•
𝑑𝑦2
𝑑𝑥2 = 60𝑥2
•
𝑑𝑦3
𝑑𝑥3 = 120𝑥2−1
=
𝑑𝑦3
𝑑𝑥3 = 120𝑥
•
𝑑𝑦4
𝑑𝑥4 = 120 =
𝑑𝑦5
𝑑𝑥5 = 0

8. Product rule
• F(x)= (𝑥2
−5)(𝑥 − 𝑥3
)
• F(x)= U.V
• U = (𝑥2
−5) V= (𝑥 − 𝑥3
)
•
𝑑𝑦
𝑑𝑥
=UV’+VU’
•
𝑑𝑦
𝑑𝑥
= (𝑥2
−5)(1-3𝑥2
) + (𝑥 − 𝑥3
)2𝑥
•
𝑑𝑦
𝑑𝑥
= 𝑥2
-3𝑥4
-5+15𝑥2
+2𝑥2
-2𝑥4
•
𝑑𝑦
𝑑𝑥
= -5𝑥4
+18𝑥2
-5

9. Quotient rule
• 𝑌 =
𝑈
𝑉
• 𝑌 =
3𝑥2+7
5𝑥+10
U = 3𝑥2
+ 7 V = 5𝑥 + 10
•
𝑑𝑦
𝑑𝑥
=
𝑈′ 𝑉−𝑉′ 𝑈
𝑉2
•
𝑑𝑦
𝑑𝑥
=
6𝑥 5𝑥+10 −5(3𝑥2+7)
(5𝑥+10) 2
•
𝑑𝑦
𝑑𝑥
=
30𝑥2+60𝑥−15𝑥2−35)
(5𝑥+10) 2
•
𝑑𝑦
𝑑𝑥
=
15𝑥2+60𝑥−35)
(5𝑥+10) 2

10. The End