Differentiation

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Differentiation

• Derivatives

• By Doug Robeson

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What is differentiation for?

• Finding the slope of a tangent line
• Finding maximums and minimums
• Finding the shape of a curve
• Finding rates of change and average rates of
  change
• Physics, Economics, Engineering, and many other
  areas of study

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Definitions of the Derivative

• The slope of a tangent line to a function
• Change in y over the change in x dy/dx
• The limit definition
• lim f(x h) - f(x) h?0
  h

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A Tangent Line
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Basic Derivative Rules

• Power Rule d(xn) nx(n-1)
• Constant Rule d(c) 0
• Note u and v are functions
• Product Rule d(uv)uv vu
• Quotient Rule d(u/v)vu - uv
  v²
• Chain Rule d(f(g(x))) f(g(x))g(x)

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Examples of Derivatives

• d(x3) 3x2
• d(4) 0
• d((x5)(3x-4)) (x5)(3) (3x-4)(1)
  3x 15 3x 4
  6x 11
• d((2x² 4)³) 3(2x² 4)²(4x)
  12x(2x² 4)²

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More Examples

• d((3x4)/x²) x²(3) (3x4)(2x)
  (x²)² 3x² -
  6x² - 8x x4
  -3x 8
  x³

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Applications of Derivatives

• Finding tangent lines
• Finding relative maxes and mins

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Finding Tangent Lines

• The derivative is the equation for finding
  tangent slopes to a function
• To find the tangent line to a function at a
  point
• Take derivative
• Plug in x value (this gives you slope)
• Put slope and point into point slope form of the
  equation of a line

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Example

• Find the tangent line to y 3x³ 5x² - 9 when x
  1.
• dy/dx 9x² 10x
• slope 9(1)² 10(1) 9 10 -1
• Have slope, need point y 3(1)³ 5(1)² - 9
  -1point (1,-1) slope -1y (-1) (-1)(x
  (1))y 1 -x 1y -x is the tangent line to
  the original function at x 1.

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Finding Relative Maxes or Mins

• The derivative is the easiest way to find the
  maximum or minimum value of a function.
• Take the derivative
• Set the derivative equal to 0
• Solve for x
• Take the derivative of the derivative (2nd
  derivative)
• Plug x values in 2nd derivative
• If positive, minimum if negative, maximum

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Example

• Find the relative maxes and/or mins of y x² -
  4x 7.
• dy/dx 2x 4
• 2x 4 0
• x2
• Second derivative (d²y/dx²) 2
• Plugging anything into d²y/dx² and its positive,
  so x2 is a relative minimum.

Back to applications
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For more practice with Derivatives

• Homework
• Page 125, 1-53 odd
• Find a web page that talks about derivatives in
  some way, write it down and a brief description
  of the page.

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End of Showby Doug Robeson