Review of Derivatives
Dan Sheldon
November 18, 2014
Motivation
Functions of one or more variables
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f (x) = (5x − 4)2
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g(x, y) = 4x2 − xy + 2y 2 − x − y
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Optimization: find inputs that lead to smallest (or largest)
outputs
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value of x with smallest f (x)
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(x, y) pair with smallest g(x, y)
Derivative
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Function f : R → R
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Derivative
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(Also f 0 (x), but we usually prefer the other notation)
d
dx f (x)
Interpretation
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Slope of tangent line at x
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Illustration: function, tangent line, rise over run
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Rate of change
f (a + ) ≈ f (a) + d
f (a)
dx
Optimization!
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If x is a maximum or minimum of f , then the derivative is zero
d
f (x) = 0
dx
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Illustration: minimum, maximum, inflection point
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So, one way to find maximum or minimum is to set the
derivative equal to zero and solve the resulting equaiton for x
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Need an expression for
d
dx f (x)
Rules
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Polynomial:
d k
x = kxk−1
dx
Rules
d k
x = kxk−1
dx
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Polynomial:
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Scalar times function:
d
d
(af (x)) = a f (x)
dx
dx
Rules
d k
x = kxk−1
dx
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Polynomial:
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Scalar times function:
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Addition:
d
d
(af (x)) = a f (x)
dx
dx
d
d
d
(f (x) + g(x)) =
f (x) +
g(x)
dx
dx
dx
Rules
d k
x = kxk−1
dx
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Polynomial:
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Scalar times function:
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Addition:
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Chain rule
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d
d
(af (x)) = a f (x)
dx
dx
d
d
d
(f (x) + g(x)) =
f (x) +
g(x)
dx
dx
dx
f (g(x))0 = f 0 (g(x)) · g 0 (x)
df
df dg
=
·
dx
dg dx
d
df
d
f (g(x)) =
·
g(x)
dx
dg(x) dx
Chain rule example
d
(5x − 4)2 =
dx
Chain rule example
d
d
(5x − 4)
(5x − 4)2 = 2 · (5x − 4) ·
dx
dx
Rules
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d
dx
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d x
dx e
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Quotient rule, product rule, etc.
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Many good references online
log x =
1
x
= ex
Example
d 3
4x =
dx
Example
d
d 3
4x = 4 x3 (scalar times function)
dx
dx
Example
d
d 3
4x = 4 x3 (scalar times function)
dx
dx
= 4 · 3x2
(polynomial)
Example
d
d 3
4x = 4 x3 (scalar times function)
dx
dx
= 4 · 3x2
(polynomial)
= 12x2
Example
d
(5x − 4)2 =
dx
Example
d
d
(5x − 4)
(5x − 4)2 = 2 · (5x − 4) ·
dx
dx
(chain rule)
Example
d
d
(5x − 4)
(5x − 4)2 = 2 · (5x − 4) ·
dx
dx
d
d
= 2 · (5x − 4) · ( 5x −
4)
dx
dx
(chain rule)
(addition)
Example
d
d
(5x − 4)
(chain rule)
(5x − 4)2 = 2 · (5x − 4) ·
dx
dx
d
d
= 2 · (5x − 4) · ( 5x −
4)
(addition)
dx
dx
= 2 · (5x − 4) · (5 − 0)
(polynomial)
Example
d
d
(5x − 4)
(chain rule)
(5x − 4)2 = 2 · (5x − 4) ·
dx
dx
d
d
= 2 · (5x − 4) · ( 5x −
4)
(addition)
dx
dx
= 2 · (5x − 4) · (5 − 0)
(polynomial)
= 10 · (5x − 4)
= 50x − 40
Exercises
Take derivative of same function, but first multiply out the
quadratic:
d
(5x − 4)2 =
dx
Back to optimization
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Our function and derivative:
f (x) = (5x − 4)2
=⇒
d
f (x) = 50x − 40
dx
Back to optimization
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Our function and derivative:
f (x) = (5x − 4)2
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=⇒
d
f (x) = 50x − 40
dx
Set equal to zero:
0=
d
f (x) = 50x − 40
dx
Back to optimization
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Our function and derivative:
f (x) = (5x − 4)2
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d
f (x) = 50x − 40
dx
Set equal to zero:
0=
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=⇒
d
f (x) = 50x − 40
dx
Solve:
x = 4/5
Back to optimization
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Our function and derivative:
f (x) = (5x − 4)2
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d
f (x) = 50x − 40
dx
Set equal to zero:
0=
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=⇒
d
f (x) = 50x − 40
dx
Solve:
x = 4/5
Convex functions
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Is x = 4/5 a minimum, maximum, or inflection point?
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Illustration: convex / concave functions
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Second derivative
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Convex = bowl-shaped
d2
f (x)
dx2
:=
d
d
dx dx f (x))
A function is convex if
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d
dx f (a)
= f 00 (x)
d2
f (x)
dx2
≥ 0 for all x
= 0 implies that a is a minimum
Wolfram Alpha
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Wolfram Alpha: http://www.wolframalpha.com/
(Optional Exercises)
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d √
dx x
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d
4x
dx 3e
Wrap-up
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What to know
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Intuition of derivative
How to take derivatives of simple functions
Convex, concave
Find minimum by setting derivative equal to zero and solving
(for convex functions)
Resources
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Lots of material online
Wolfram Alpha: http://www.wolframalpha.com/
Mathematica, Maple, etc.