Differentiation

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Differentiation

Session 3Business Mathematics1The syllabus that this topic covers deal with calculus only at an elementary level and require:familiarity with a few simple rulesan understanding of some relevant applications

Business MathematicsSlide 2 of 29

Rule 1: The derivative of a constant is zero.Example:Let Y = 7Thus:dy/dx = 0yX07Rules of calculusBusiness Mathematics3Rule 2: The derivative of a constant times a variable is simply the constant.Example:Let y = 13xThus:dy/dx = 13yx1326210Rule 2Business Mathematics4

Rule 3: A power function has the form y = axn, where a and n are constants. The derivative of a power function is:Example:Let y = 4x3

Thus:

Business Mathematics5Rule 3

Special cases of the power function Note the following:y =1/x2 is equivalently written as y = x-2and

can be written y = x1/2Hence by rule 3 (or the power rule), the respective derivatives are given by:dy/dx = -2x-3Anddy/dx = .5x-1/2 =

Power functionsBusiness Mathematics6Rule 4: Suppose the product of two functions : y = f(x)g(x). Then we have:

Example:Let y = (4x)(3x2)Thus:dy/dx = (4)(3x2) + (6x)(4x) = 36x2Rule 4Business Mathematics7Rule 5Business Mathematics8 Rule 5: The derivative of the sum of functions is equal to the sum of the derivatives.

If y = f(x) + g(x), then:dy/dx = df/dx + dg/dxExample: Let: y = .1x2 2x3 Thus: dy/dx = .2x 6x2

Rule 6: Suppose y is a quotient: y = f(x)/g(x). Then we have:

ExampleSuppose we have: y = x/(8 + x)Thus:dy/dx = [1 (8 + x) 1 (x)]/ (8 + x)2 = 8/(8 + x)2 Rule 6Business Mathematics9The marginal profit (M) functionBusiness Mathematics10Let the profit function be given by: = 2Q - .1Q2 3.6To obtain the marginal profit function, we take the first derivative of profit with respect to output (Q): M = d/dQ = 2 - .2QTo solve for the output level that maximizes profits, set M =0. 2 - .2Q = 0 Thus: Q = 10Differentiation Differentiation can be thought of as a process which transforms one function into a different one. The new function is known as the derivative of the original one.The derivative of the simple function,y = axb is given by Business MathematicsSlide 11 of 29

The derivative of the sum of two or more simple functions is the sum of the separate derivatives of the functions.Differentiation can be repeated as many times as necessary on any given function.If y is any function of x:


Solving the equation, will identify the turning points of a function. The second derivative can then be used to indicate whether the turning point is a maximum or minimum.when the second derivative is positive, it signifies a minima.when the second derivative is negative, it signifies a maxima.Business MathematicsSlide 13 of 29

Use of differentiation in a business contextIn any commercial environment, the profit can be considered as a simple function of the difference between the revenue obtained from the sale of a number of products and the costs involved in their production.

Profit = Revenue - CostBusiness MathematicsSlide 14 of 29Cost functionThe cost involved in standard processes can be normally categorised as follows:fixed costvariable costspecial/optional costTotal cost = fixed cost + variable cost + special cost

Business MathematicsSlide 15 of 29Average cost =

Marginal cost, is defined as the extra cost incurred of producing another item at activity level x.Business MathematicsSlide 16 of 29

Cost maximisation or minimisation occur when = 0.

The second differential coefficient is used to determine whether there is a maximum or minimum.Business MathematicsSlide 17 of 29

Revenue functionR(x) = x . Pr (x)where x is the quantity of items demanded (supplied or produced). Pr (x) is the fixed cost associated with the product.

Business MathematicsSlide 18 of 29Price /demand functionIn business environment, item price is usually dependent on the number of items in demand.The more items that are in demand, the less the price per unit is.Business MathematicsSlide 19 of 29Procedure for obtaining the demand functionput the demand function in the form Pr = a + bxsubstitute the first price/ demand pair into Pr to obtain one equation in a and bsubstitute the second price/demand pair into Pr to obtain another equation in a and b.Solve the two equations obtained, for a and b.Business MathematicsSlide 20 of 29Marginal revenue, is the extra revenue obtained from producing another item at activity level x.

Revenue maximisation or minimisation occur when = 0.

The second differential coefficient is used to determine whether there is a maximum or minimum.


Profit functiondifference between revenue function and cost function, P = Revenue costProfit maximisation or minimisation occur when = 0.The maximum profit point for some process can be found by solving the equation:


Break-even pointIt is the level of production where profit is zero or when the total revenue for a period will equal the total costs for the same period. i.e. R = CHence, Pq = F + Vq

where q is the quantity of units p is the selling price F is the fixed cost V is the variable cost


Break-even chartIt is a graphical representation of the cost and revenue associated with different levels of output. It is useful in business decision-making as it enables companies to identify the level of output which, given the prevailing cost and revenue function, maximises profit.Business MathematicsSlide 24 of 29Optimization using CalculusBusiness Mathematics25

We will review some rules of differential calculus that are especially useful for management decision making

Suppose that a business firm has estimated its profit () function (based on marketing and production studies) as follows:

Where is profit (in thousands of dollars) and Q is quantity (in thousands of units). Thus the problem for management is to set its quantity(Q) at the level that maximizes profits (). The profit functionBusiness Mathematics26

The profit function shows the relationship between the managers decision variable (Q) and her objective (). That is why we call it the objective function.What is an objective function?Business Mathematics27

Marginal profit at a particular output is given by the slope of a line tangent to the profit functionThe profit function againBusiness Mathematics28

Computing profit at various output levels using a spreadsheetBusiness Mathematics29

Recall our profit function is given by:

= 2Q - .1Q2 3.6Fill in the quantity column with 0, 2, 4, . . .Assume that you typed zero in column cell a3 of your spreadsheetPlace your cursor in the the cell b3 (it now contains the bolded number 3.6)just to the right of cell a3. Type the following in the formula bar:

=(2*a3)-(.1*a3^2)-3.6

and click on the check mark to the left of the formula bar.Now move your cursor to the southeast corner of cell b3 until you see a small cross (+).Now move your cursor down through cells b4, b5, b6 . . . to compute profit at various levels of output. Chart1-3.602.84.866.464.82.80-3.6

QuantityProfitThe Profit Function

Sheet1QuantityProfit(000s)(000s)0.0-3.62.00.04.02.86.04.88.06.010.06.412.06.014.04.816.02.818.00.020.0-3.6

Sheet2

Sheet2-3.602.84.866.464.82.80-3.6

QuantityProfitThe Profit Function

Sheet3

Sheet1QuantityProfit(000s)(000s)0.0-3.62.00.04.02.86.04.88.06.010.06.412.06.014.04.816.02.818.00.020.0-3.6

Sheet2

Sheet3

Differentiation

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