Derivatives for B. Com

1. B.Com First Year Math
Suraj – 7709685533
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2. Derivatives of Standard Function
We have studied Constant Function , Power Function ,
Demand function , Revenue function ,cost Function
Y is dependent variable and depends on X
Y – Price X =Demand
Price change by 100 , Demand changes by 10
Rate of Change of Y with respect to X is called Derivatives
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3. Alternative Notation
• Let 𝑓 𝑥 𝑏𝑒 𝑎 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑏𝑙𝑒 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑥 𝑎𝑛𝑑 𝑌 = 𝐹 𝑥
• 𝑓′ 𝑥 , 𝑡ℎ𝑒 𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑜𝑓 𝑓 𝑥 𝑤. 𝑟. 𝑡. 𝑥 𝑎𝑡 𝑥 , 𝑒𝑥𝑖𝑠𝑡𝑠
• 𝑓′
𝑥 𝑖𝑠 𝑎𝑙𝑠𝑜 𝑑𝑒𝑛𝑜𝑡𝑒𝑑 𝑏𝑦
𝑑𝑦
𝑑𝑥
or
𝑑 𝑦
𝑑𝑥
• If 𝑦 = 𝑓 𝑥 𝑖𝑠 𝑎 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑏𝑙𝑒 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑥 𝑡ℎ𝑒𝑛
𝑑𝑦
𝑑𝑥
= 𝑓′(𝑥)
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4. Algebra of Derivatives
1. y= u+V
𝑑𝑦
𝑑𝑥
=
𝑑𝑢
𝑑𝑥
+
𝑑𝑣
𝑑𝑥
dy =
y2- y1
2. Y= U-V
𝑑𝑦
𝑑𝑥
=
𝑑𝑢
𝑑𝑥
-
𝑑𝑣
𝑑𝑥
3. Y=CU
𝑑𝑦
𝑑𝑥
= 𝐶
𝑑𝑢
𝑑𝑥
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5. 1. Y= UV
𝑑𝑦
𝑑𝑥
= 𝑈
𝑑𝑣
𝑑𝑥
+ V
𝑑𝑢
𝑑𝑥
1 2 1 2
1. Y= U/V dy/dx = Vdu/dx – U dv/dx
𝑉2
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6. Derivatives of Standard Functions
1. If y=C where C is a fixed real number , then
𝑑𝑦
𝑑𝑥
= 0
y = 5 , y= 6 e.g Price constant of any product
1. If 𝑦 = 𝑥 𝑛 where n is a fixed real number then
𝑑𝑦
𝑑𝑥
= 𝑛𝑥 𝑛−1
𝑦 = 𝑥2 , 𝑦 = 𝑥6, 𝑦 = 𝑥8
1. If 𝑦 = 𝑒 𝑥 , then
𝑑𝑦
𝑑𝑥
= 𝑒 𝑥
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7. 1. If 𝑦 = 𝑎 𝑥
where a is a fixed positive real numer like ( 1,2,3,4 )
𝑑𝑦
𝑑𝑥
= 𝑎 𝑥
𝑙𝑜𝑔𝑎
5. If 𝑦 = 𝑙𝑜𝑔𝑥 where x>0 , then
𝑑𝑦
𝑑𝑥
=
1
𝑥
6. If y= x
𝑑𝑦
𝑑𝑥
= 1
7. If𝑦 = 𝑥−
1
2 then
𝑑𝑦
𝑑𝑥
=
8. If𝑦 = 𝑥
1
2 then
𝑑𝑦
𝑑𝑥
=
Remark : when we write logx without specifically writing the base then
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8. Illustration 1
1. 𝑋4 + 𝑒 𝑥 + 5 𝑥 − 𝑙𝑜𝑔𝑥 + 7
2. 4𝑥7 − 5𝑒 𝑥 + 4𝑋3 𝑥 − 10 𝑙𝑜𝑔𝑥 − 25
3. 7 𝑥 − 2𝑥
7
2 + 3𝑙𝑜𝑔𝑥 − 𝑥 + 𝑥−
1
2
4. 𝑦 = 𝑥2
+ 1 𝑥 − 5
5. 𝑦 = 𝑥 − 4 (𝑥2 + 3 )
6. 𝑦 = 𝑎 𝑥
𝑥 𝑎
7. 2𝑥2 − 3𝑥 + 4
1/2
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9. 1. 𝑦 =
𝑥+2
2𝑥−3
2. 𝑦 =
𝑥+𝑙𝑜𝑔𝑥
𝑥+1
3. 𝑦 = 𝑥𝑒 𝑥 + 𝑙𝑜𝑔𝑥
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10. Significance of the sign of derivative
We shall now state two results without proofs
1. If f is a differentiable at x=a and f’(a)>0 then f is increasing at x=a
2. If f is a differentiable at x=a and f’(a)<0 then f is decreasing at x=a
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11. Derivatives of Second order
• Let y= f(x) be a differentiable function of x
• Dy/dx = f’(x)
𝑦 = 𝑓 𝑥 = 𝑥2 + 𝑒3𝑥
𝑑𝑦
𝑑𝑥
= 𝑓′ 𝑥 = 2𝑥 + 3𝑒3𝑥
𝑑2 𝑦
𝑑2 𝑥
=𝑓′′ 𝑥 = 2 + 9𝑒3𝑥
Solve 𝑓 𝑥 = 𝑥3 − 9𝑥2 + 24𝑥 + 100
Find 𝑓′ 𝑥 𝑎𝑛𝑑 𝑓′′ 𝑥
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12. Maxima and Minima
1. If f’(a) =0 and f’’(a) <0 , then f has a maximum at x= a
2. If f’(a) =0 and f’’(a) >0 , then f has a minimum at x= a
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13. Applications of Derivatives to Economics
,commerce and Management
1. Marginal Cost Function
2. Marginal Revenue Function
3. Elasticity
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14. Marginal Cost Function
• C is the Total cost of Function for producing x Units
• The rate of change of cost w.r.t the number of units produced is the
Marginal Cost and is denoted By MC
• The Marginal Cost is Given By
𝑑𝑐
𝑑𝑥
𝑀𝐶 =
𝑑𝑐
𝑑𝑥
• The rate of change of average cost w.r.t the number of units produced
is called Marginal Average Cost and is denoted By MAC
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15. Marginal Revenue Function
• If D Units of a commodity are demanded then D is called Demand .
If P is the selling price of each unit and R is the total revenue then R=pD
R is called the total Revenue function
A relation Between p and D is called the Demand function
The Rate of Change of total revenue w.r.t the Demand is called Marginal
Revenue and is denoted by MR
𝑀𝑅 =
𝑑𝑅
𝑑𝐷
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16. Elasticity
• Let D be the demand and p be the price .
• The quantity
−
𝑝
𝐷
dD
dp
is called the elasticity of demand w. r. t price and is denoted by 𝞰
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17. Illustartion 12
• The cost of manufacturing x items of a product is given by 2𝑥2 +
3𝑥 +
10 . 𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑡𝑜𝑡𝑎𝑙 𝑐𝑜𝑠𝑡 , 𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑐𝑜𝑠𝑡 , 𝑚𝑎𝑟𝑔𝑖𝑛𝑎𝑙 𝑐𝑜𝑠𝑡 𝑎𝑛𝑑 𝑡ℎ𝑒
marginal average cost if 10 items are manufactured .
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