The Limit Definition of the Derivative, Part 2

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• Misc
  • Sign up to grade problem set 2
• Questions?
  • Graphing in R
    • 3 basic steps:
      • Make list of x values
      • Make a list of correspomding y values
      • Use plot function to draw graph (w/ type=“l”)
    • For example, plot x⁴ + 6 over -10 ≤ x ≤ 10
      

      > x = seq( -10, 10, by=0.02 )
      > y = x^4 + 6
      > plot( x, y, type="l" )

      
      
    • (There is also a curve function that is somewhat easier to use—type help( curve ) in R for more information)
• Derivatives as Limits: Examples
  • f′(x) = lim_h→0( f(x+h) - f(x) ) / h
  • f(x) = 1/x³
    
  • f(x) = √(2x)
    • Conjugate
      • (√a - √b) (√a + √b) = (a - b)
    • Apply to √(2x+2h) - √(2x)
      
    • How would you do this with power rule?
      • Needs chain rule too (so this is beyond what we’re doing so far)
      • d/dx (2x)^1/2
      • = 1/2 (2x)^-1/2 (2)  power rule + chain rule
      • = (2x)^-1/2
      • = 1 / √(2x)
• Basic Differentiation Rules
  • Any examples you know of?
    • if f(x) = k g(x) for any constant k, then f′(x) = k g′(x)
    • if f(x) = xⁿ then f′(x) = n x^n-1
    • if f(x) = g(x) h(x) then f′(x) = g′(x)h(x) + g(x)h′(x)
    • if f(x) = g(x) / h(x) then f′(x) = ( g′(x)h(x) ) - (h′(x)g(x) ) / h²(x)
    • if f(x) = g (h(x) ) then f′(x) = g′( h(x) ) h′(x)
  • How does if f(x) = k g(x) for any constant k, then f′(x) = k g′(x) follow from limit definition?
    • f′(x) = lim_h→0( f(x+h) - f(x) ) / h
    • = lim_h→0( kg(x+h) - kg(x) ) / h
    • = lim_h→0 k ( g(x+h) - g(x) ) / h
    • = k lim_h→0( g(x+h) - g(x) ) / h
    • = k g′(x)
• Next
  • “Powers, Multiples, Sums, and Differences” part of section 3.3
    • • Middle of page 118 - bottom of 121

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