DIFFERENTIATION

Differentiation is the process of finding the derivative of a function. The derivative

represents the instantaneous rate of change in the dependent variable given a change in the
independent variable.

Rules of differentiation
(a) Power Rule
dy
If y = x n , = nx n −1 . This holds at all points except at x = 0
dx
(b) Product Rule
If y is a product of two functions i.e. y = uv where u and v are functions of x , then
dy dv du
=u +v
dx dx dx
Examples
1. Find the derivative of the function y = 2 x 2 (5 x + 3)
2. Find the derivative of the function y = (x + 5)(x 2 + 3)
(c) Quotient Rule
u
If y is a quotient of two functions i.e. y = where u and v are functions of x , then
v
du dv
v −u
dy dx dx
=
dx v2
Examples

1. Find the derivative of the function

2. Find the derivative of the function

(d) Chain rule
Chain rule is used to differentiate compound functions. If y = (u ) where u is a function
n

dy dy du
of x , then = *
dx du dx
Examples

1. Find the derivative of the function y = x 3 − 5 x + 3

2. Find the derivative of the function y = (3 x 2 + 6 x − 5)
10

Second derivative of a function

 d2y
The second derivative of y = f (x) is denoted by f ' ' ( x) or .
dx 2
Example
Find the second derivative of the function 𝑦 = 5𝑥 3 + 4𝑥 2 + 10𝑥 + 25
Finding a stationary point
A stationary point is that point where the gradient is zero. It may be a maximum or a
minimum point. The following steps are taken to find out a stationary point
➢ Find the derivative of the function
➢ Equate the derivative to Zero
➢ Solve the resulting equation
Example
Find the point at which the curve Y = 2 x 2 − 3x − 10 will have a stationary point.

Turning points (Maximum and Minimum)

The turning points show the maximum or minimum values of a given function. The
gradient at both maximum and minimum points is zero. In order to find out the maximum
or minimum points, the second derivative is calculated. The following rules apply:
First derivative Second derivative
Maximum Zero Negative
Minimum Zero Positive

Example
Find the point at which the curve Y = x 3 − 3x + 2 will have a maximum or minimum point.

Application of Differentiation to Revenue, Cost and Profit Functions

Revenue function
Total Revenue = (Price per unit) (Quantity sold) = P*Q
Total Revenue
Average Revenue (AR) =
Quantity
dTR
Marginal Revenue (MR) =
dQ
Marginal Revenue is the additional revenue derived from selling one more unit of a product
or service. If each unit of a product sells at the same price, the marginal revenue is always
equal to the price.
𝑑𝑇𝑅 𝑑2 𝑇𝑅
Total revenue would be maximized when = 0 and <0
𝑑𝑞 𝑑𝑞 2
Illustration 1:
The demand for the product of a firm varies with the price that the firm charges for the
product. The firm estimates that annual total revenue R (stated in 1000” s) as a function of
the price P is given by
a. Determine the price, which should be charged in order to maximize total revenue.
b. What is the maximum value of annual total revenue?

Illustration 2
A public transportation company has been experimenting on a possibility of developing a
system of charging fares. The demand functions, which expresses the ridership as a
function of fare charged is given below:

where equals the average number of riders per hour and equals the fare in shillings.
a. Determine the fare, which should be charged in order to maximize hourly bus fare
revenue.
b. What is the expected maximum revenue?
c. How many riders per hour are expected under this figure?

Cost Function
Total Cost (TC) = Fixed Cost + Total Variable Cost
Total variable cost = unit variable cost * Quantity
Total Cost
Average cost (AC) =
Quantity
dTC
Marginal Cost (MC) =
𝑑𝑄

Fixed costs are costs, which do not change with the level of production while variable costs
are costs, which change with a change in the level of production. Marginal cost is the
additional cost incurred as a result of producing one more unit of a product or service.
𝑑𝑇𝐶 𝑑2 𝑇𝐶
Total cost would be minimized when = 0and >0
𝑑𝑞 𝑑𝑞 2

Example:
A retailer of motorized bicycles has examined cost data and has determined a cost function
which expresses the annual cost of purchasing, owning, and maintaining inventory as a
function of the size (number of units) of each order it places for the bicycles. The cost of
function is,

Where equals annual inventory cost, stated in dollars and equals the number of cycles
ordered each time the retailer replenishes the supply.
a. Determine the order size, which minimizes annual inventory cost.
b. What is minimum annual inventory cost expected to equal?

Profit Function

Profit = Total Revenue - Total Cost

dP d 2P
Total profit would be maximized when = 0 and 0
dq dq 2
Example 1:
A major cosmetic and beauty supply firm, which specializes in door-to-door sales
approach, has found that the response of sales to the allocation of additional sales
representation behaves according to the law of diminishing returns .For one region sales
district, the company estimates that annual profit , stated in hundreds of shillings, is a
function of the number of sales representations assigned to the district .The function
these two variables is

a. What number of representations will result in maximum profit for the district?
b. What is the expected maximum profit?

Example 2:
A manufacturer has developed a new design for solar collection panels. Marketing studies
have indicated that annual demand for the panels will depend on the price charged. The
demand function for the panels has been estimated as

Where equals the number of units demanded each year and equals the price in
shillings. Engineering studies indicate that the total cost of producing panels is estimated
well by the function . Formulate the profit function p = f (q)
which states the annual profit as a function of the number of units which are produced
and sold.

Integral Calculus
Integration is the reverse of differentiation. Given the derivative of a function, the process
of finding the original function is the reverse of differentiation.

Power Rule:
𝑥 𝑛+1
∫ 𝑥 𝑛 𝑑𝑥 = 𝑛+1
+ 𝐶 Provided that 𝑛 ≠ −1

Power Rule Exception

 ∫ 𝑥 −1 𝑑𝑥 = 𝑙𝑛 𝑥 + 𝐶

Application of integration to cost, revenue and profit functions

1. The function describing the marginal cost of producing a product is 𝑀𝐶 = 𝑥 + 100
Where x equals the number of units produced .It is also known that total cost equals
Sh. 40,000 when x = 100 . Determine the total cost function.

2. The marginal revenue function for a company’s product is

𝑀𝑅 = 50,000 − 𝑥
Where x = the number of units produced and sold. If total revenue equals 0 when
no units are sold, determine the total revenue function for the producer.

3. An automobile manufacturer estimates that the annual rate of expenditure r (t ) for

maintenance on one of its model is represented by the function r (t ) = 1000 + 100t 2 .
i. What is the expected maintenance expenditures during the automobile first 5
years?
ii. Of the above expenditures, what is expected to be incurred during the fifth
year?

APPLICATIONS OF CALCULUS
1. A manufacturer has found that if he wants to increase his output, he must lower his
price. His total revenue (TR) from an output x, is given by the expression𝑇𝑅 =
𝑥(148 − 𝑥). His production costs are Shs. 1,000 fixed and Shs. 36 per unit variable.
Required:
Find:
i. The output that would maximize revenue.
ii. The maximum total revenue.
iii. The profit P in terms of the number of units (x).
iv. The output x, that would maximize profit.

2. Given f ( x) = x 3 − 6 x 2 + 9 x + 1 , find the maximum and the minimum value of f (x)

3. A car costs Sh.75, 000 in the market and the running cost for x kilometers is given by
𝑉. 𝐶(𝑥) = 𝑥 + 30𝑥(𝑥 − 1). Find, when is the total average cost minimum.

4. A factory produces x calculators per day. The total daily cost in shillings incurred is
5x 2 − 800x + 500. if the calculators are sold for Sh. (100 –10x) each, find the number
of calculators that would maximize the daily profit.
5. The cost accountant of a firm producing colour television has worked out the total cost
function for the firm as 𝑇𝐶 = 120𝑄 − 𝑄 2 + 0.02𝑄 3 . A sales manager has provided
the sales forecasting function as 𝑃 = 114 − 0.25𝑄 where P is price and Q the
quantity sold. Required: -
1. Find the level of production that will yield minimum average cost per unit
and determine whether this level of output maximizes profit for the firm.
2. Determine the price that will maximize profit for the firm.
3. Determine the maximum revenue for this firm.

6. A firm has analyzed their operating conditions, prices and costs and have developed
the following functions:
Revenue = 400𝑄 − 4𝑄 2 and cost = 𝑄 2 + 10𝑄 + 30 where Q is the number of units
sold. The firm wishes to maximize profit and wishes to know:
➢ What quantity should be sold?
➢ At what price?
➢ What will be the amount of profit?
➢ Find the point of maximum value of the revenue function?

7. A firm has analyzed its operating conditions and has developed the following functions
Total Revenue = −10𝑞 2 + 200𝑞
Total Cost = 𝑞 2 − 20𝑞 + 1000
Where q is the number of units produced and sold?
Determine the value of q that:-
a) Maximizes revenue and hence the Maximum Revenue
b) Minimizes Cost and hence the Minimum Cost
c) Maximizes profit and hence the Maximum Profit.

8. The cost function of a company is given by C ( x) = x 3 − 60 x 2 + 1200x + 1000 . If the

product sells for Ksh. 588 each,
i. What level of production will maximize the profits?
ii. Find the total cost, total revenue and total profit at the optimal production.

9. The total profit per acre on a wheat farm, has been found to be related to the
expenditure per acre for (a) Labour and (b) soil conditioners and fertilizers. If x
represents the shillings per acre spent on labour and Y represents the shillings per acre
spent on soil improvement:
Profit 𝜋 = 48𝑋 + 60𝑌 + 10𝑋𝑌 − 10𝑋 2 − 6𝑌 2
What are the values of X and Y that maximize profits?
10. A company sells two products. Total revenue from the two products is estimated to be
a function of the numbers of units sold of the two products. Specifically, the function
is R = 30000x + 15000y − 10 x 2 − 10 y 2 − 10 xy where R equals total revenue and x and
y equal the number of units sold of the two products. Determine the number of each
product that should be produced in order to maximize total revenue and hence the
maximum revenue.

11. ABC Ltd employed a cost accountant who developed two functions to describe the
operations of the firm. He found the marginal revenue function to be 𝑀𝑅 = 25 − 5𝑥 −
2𝑥 2 and the marginal cost function to be 𝑀𝐶 = 15 − 2𝑥 − 𝑥 2 where x is the level of
output. Determine the profit maximizing output and the total profit at that point.
12. Given Marginal Cost (MC) = 36 + (q – 8)2 and Marginal Revenue (MR) = 100 – 2q
where q is quantity of units. Find: -
➢ The profit maximizing output of the firm.
➢ The total cost and total revenue function assuming that fixed cost and fixed
revenue is equal to zero.
➢ The value of maximum profit.

13. The total cost and total revenue function for a product are
𝐶(𝑞) = 500 + 100𝑞 + 0.5𝑞 2 and 𝑅(𝑞) = 500𝑞
➢ Using the marginal approach, determine the profit-maximizing level of output.
➢ What is the maximum profit?

14. The total cost and total revenue functions for a product are
𝐶(𝑞) = 40,000 + 25𝑞 + 0.025𝑞 2 and 𝑅(𝑞) = 75𝑞 − 0.008𝑞 2
i.Using the marginal approach, determine the profit-maximizing level of output.
ii.What is the maximum profit?