mathematicalmarketing slide 3a.1 mathematical tools chapter 3: mathematical tools we will be...
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Slide 3a.Slide 3a.11Mathematical ToolsMathematical Tools
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Chapter 3: Mathematical Tools
We will be reviewing
Exponents and Logarithms.
Scalar Calculus
Partial Derivatives of a Scalar Function of a Vector
Slide 3a.Slide 3a.22Mathematical ToolsMathematical Tools
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The Ln Function
ln ab = ln a + ln b
ln (a/b) = ln a – ln b
ln ab = b ln a
ln ea = a
ln e = 1
ln 1 = 0
ln a = c if
ec = a
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Rules for Exponents
ab · ac = ab+c
a½ = √a
a-1 = 1/a
We often say exp(a) instead of ea.
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The Derivative Is the Limit of a Slope at a Point
x x + x
x2
(x + x)2
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Rise Over Run
.xx2
x
x)x(xx2x
x)xx(
x)xx(slope
222
22
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Some Rules for Scalar Derivatives
.xmdx
dx 1mm
d(c)/dx = 0
f(x) = xm
f(x) = c
f(x) = cx
f(x) = x2 dx2/dx = 2x
dcx/dx = c
Derivative Function
xx
edx
def(x) = exp(x)
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Compound Functions
dx
)x(dg
dx
)x(df
dx
)]x(g)x(f[d
.dx
)x(df
)x(df
)]x(f[dg
dx
)]x(f[dg
dx
)x(dfe
dx
de )x(f
)x(f
The derivative of a sum
The derivative of a function of a function
The derivative of afunction of an exponent
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Partial Derivatives
m21 x
)'(f
x
)'(f
x
)'(f)'(f xxx
x
x
By definition
.]000[
x
c
x
c
x
cc
m21
x
A very simple example:
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The Derivative of a Linear Combination
m21 xxx
xaxaxa
x
xa
]xaxaxa[x
]xaxaxa[x
]xaxaxa[x mm2211
m
mm2211
2
mm2211
1
.aaa m21 a
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The Derivative of a Quadratic Form
Axx
Axx
2
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The Transpose of a Derivative Is the Derivative of the Transpose
xx
xx
ff
andff
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Second Order Derivatives
.22
ijjiij xx
f
xx
f
x
f
x