the chain rule

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the chain rule
the greatest thing to happen to calculus since
differentiation
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what is


the chain rule
Dx(f(g(x))) f (g(x))g(x)
or
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what is


the chain rule

• The derivative of a composite function
  Dx(f(g(x))) is the derivative of the outer
  function in terms of the inner functionf
  (g(x), multiplied by the derivative of the
  inner function g(x).
• Sounds complicated this way
• Thats how they say it in the calc book.

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what is


the chain rule

• The derivative of a composite function where the
  inner function can be substituted with the term
  u, is equal to the differentiation of the outer
  function in terms of u multiplied by the
  differentiation of the inner function in terms of
  x.
• Sounds complicated this way
• Thats how they say it in the calc book.

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who uses


the chain rule

• 67 of Americans prefer the chain rule over any
  other method of differentiating composite
  functions.
• High school students
• College undergrads
• come on, everybodys doing it!

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when do you use


the chain rule

• Differentiating a function that can be
  represented as the composite of more than one
  function
• showing off your mathematical prowess
• whenever everyone else is doing it!

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the chain rule in


contemporary media

• Social Distortions Ball and Chain
• Aretha Franklins Chain of Fools
• Chain stores and restaurants such as McDonalds,
  The Olive Garden, Albertsons, The Gap
• Chain letters and e-mail
• Mr. T

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the chain rule


in nature

• Food Chain
• Chain Reaction
• Tire chains for driving in the snow
• Bicycle and Motorcycle chains to prevent theft

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chain rule


example
f(x) (3x2 -8x4)4 find f (x).
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chain rule


example
f(x) (3x2 -8x4)4
analysis shows g(x) 3x2 -8x4 f(x) ( g(x) )4
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chain rule


example
f(x) (3x2 -8x4)4 g(x) 3x2 -8x4 f(x) (
g(x) )4
Plug into equation
Dx(f(g(x))) f (g(x))g(x)
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f(g(x)) (3x2 -8x4)4
Dx(f(g(x))) f (g(x))g(x)
4( g(x) )3 (6x - 8)
4(3x2 -8x4) (6x - 8)
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chain rule example


afterthoughts

• Get it?
• Are you sure?
• Any questions before we bust out the big guns?
• Can you dig it?
• Do you like scary movies?
• Are you my mother?
• What is love? Baby dont hurt me no more.

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SUPER chain rule


example
f(x) sin2(5(x3)4) find f (x).
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SUPER chain rule


example
y sin2(5(x3)4)
analysis shows
s x3 t 5s4
u sin(t) y u2
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SUPER chain rule


example
y sin2(5(x3)4)
s x3 t 5s4
u sin(t) y u2

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f(x) sin2(5(x3)4)
y sin2(5(x3)4) s x3 t 5s4 u sin(t) y
u2
s 1 t 20s3 u cos(t) y 2u
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f(x) sin2(5(x3)4)
s 1 u cos(t) t 20s3 y 2u

(2u) (cos(t)) (20s3) (1)
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f(x) sin2(5(x3)4)

(2u) (cos(t)) (20s3) (1)
s x3 t 5s4 u sin(t) y u2
(2 sin(t)) (cos(5s4)) (20(x3)3)
(2 sin(5s4)) (cos(5(x3)4)) (20(x3)3)
(2 sin(5(x3)4)) (cos(5(x3)4)) (20(x3)3)
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SUPER chain rule


afterthoughts

• Get it?
• The solution was very long.
• Mr. T is my hero.
• This is the end of the presentation.

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ivanWick
rickyTaing
credits
michaelLee