lecture 05 - differentiation 1

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Advanced Mathematics for Business

Topic 5:

Differentiation 1

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Differentiation 1 Topic 5 - 5.2

Scope and Coverage

This topic will cover:• Gradient• Definition of the derivative• Rules of differentiation

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Differentiation 1 Topic 5 - 5.3

Learning Outcomes

By the end of this topic students will be able to:• Find the derivative of variables raised to a power• Use the rules of differentiation• Relate differentiation to optimization

- Obtain the economic order quantity formula

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Differentiation 1 Topic 5 - 5.4

Equations and Functions

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Differentiation 1 Topic 5 - 5.5

Linear and Non-Linear Graphsy = x

y = x2

y = ex

y = lnx

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Differentiation 1 Topic 5 - 5.6

Straight Lines (linear functions)

y

x

y = c

y = f(x) = mx + c

x = -c/m

y increases m

x increases 1

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Differentiation 1 Topic 5 - 5.7

Gradient of a Straight Line

f(x+∆x)

f(x)

x+∆xx

∆x

∆y= f(x+∆x)-f(x)

y = f(x) = mx + c

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Differentiation 1 Topic 5 - 5.8

Gradient of a Non-Linear Curve• Which line best approximates the gradient of the

non-linear curve at the point?

x xxx

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Differentiation 1 Topic 5 - 5.9

Derivative of a Non-Linear Curve

∆y= f(x+∆x)-f(x)

f(x+∆x)

f(x)

x+∆xx

∆x

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Differentiation 1 Topic 5 - 5.10

Derivative of a Quadratic

∆y= a(x+∆x)2 – x2

∆x

yy = ax2

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Differentiation 1 Topic 5 - 5.11

Derivative of an Arbitrary Power

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Differentiation 1 Topic 5 - 5.12

Examples

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Differentiation 1 Topic 5 - 5.13

Derivatives of other functions• Commonly met functions in business

- powers, exponential and natural logarithms.

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Differentiation 1 Topic 5 - 5.14

Examples

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Differentiation 1 Topic 5 - 5.15

Differentiation is Linear

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Differentiation 1 Topic 5 - 5.16

Product Rule

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Differentiation 1 Topic 5 - 5.17

Chain Rule

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Differentiation 1 Topic 5 - 5.18

Quotient Rule

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Differentiation 1 Topic 5 - 5.19

Rules of Differentiation

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Differentiation 1 Topic 5 - 5.20

Why Does this Matter? Optimisation!pr

ofit

cost

zero gradient

management decision

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Differentiation 1 Topic 5 - 5.21

Example

1 Harris (1913), Wilson (1934)

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Differentiation 1 Topic 5 - 5.22

Learning Outcomes

By the end of this topic students will be able to:• Find the derivative of variables raised to a power• Use the rules of differentiation• Relate differentiation to optimization

- Obtain the economic order quantity formula

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Differentiation 1 Topic 5 - 5.23

References• Harris, FW. (1913). How many parts to make at

once. Factory, the Magazine of Management 10(2), 135-136; reprinted in Operations Research 1990, 38(6), 947-950

• Wilson, RH. (1934). A scientific routine for stock control, Harvard Business Review 13(1), 116-128.

Differentiation 1 Topic 5 - 5.24

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Topic 5 – Differentiation 1

Any Questions?