lecture 05 - differentiation 1
TRANSCRIPT
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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
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