Basic Differentiation Rules

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• Questions?
  • curve function in R?
    • All-in-one graph plotting function
    • Basic usage is curve( expression, low, high, ...options... )
    • add=TRUE option adds new curve to existing plot
    • col="colorname" option sets color of line
    • xname="variablename" option specifies name of variable in expression to vary for plotting
      

      > curve( pi - 2/x^2, -10, 10 )
      > curve( x^3, -5, 5, add=TRUE, col="red")
      > curve( y^2, -10, 10, xname="y", add=TRUE, col="orange")

      
      
    • ylim=c(low, high) option sets range of Y values to show
      

      > curve( pi - 2/x^2, -10, 0, ylim=c(2,4))

      
      
  • Removing a variable from R?
    • rm function does it
      

      > x = 1
      > x
      [1] 1
      > rm(x)
      > x
      Error: object 'x' not found

  • RStudio is an R program with a nicer user interface
• Differentiation Rules
  • First part of section 3.3
    • Power rule for polynomials
      • d( xⁿ )/dx = n x^n-1
    • dc/dx = 0
    • Constant multiple rule: d( cu )/dx = c du/dx
      • d( 3x² )/dx = 3 dx²/dx = 3 (2x) = 6x
    • d( u + v )/dx = du/dx + dv/dx
    • Notations
      • d/dx
  • Examples
    • 5x³ - 2x + 7
      • d( 5x³ - 2x + 7 )/dx
      • = 15x² - 2
    • x √x
      • = x x^1/2
      • = x^3/2
      • d( x^3/2 )/dx = 3/2 x^1/2 = 3√x / 2
    • x² / 3√x
      • = 1/3 ( x² / √x )
      • = 1/3 ( x² / x^1/2 )
      • = 1/3 ( x^3/2 )  (algebra)
      • d( 1/3 ( x^3/2 ) )/dx
      • = √x/2 = 1/2 (x^1/2)  (power rule, constant multiple)
    • 1 / x³
      • = x^-3
      • d( x^-3 )/dx = -3x^-4  (power rule)
      • = -3/x⁴
  • Suppose f′(x) = 3x² and g′(x) = x³ - 1. What is the derivative of 3f(x) - 2g(x)?
    • Method 1, the natural approach
      • 3f(x) - 2g(x)
      • = 3x³- 2(1/4 x⁴ - x )
      • d( 3x³ - 2(1/4 x⁴ - x ) )/dx
      • = 9x² - 2x³ + 2
    • Or, the simple approach
      • d( 3f(x) - 2g(x) )/dx
      • = 3 f′(x) - 2 g′(x)
      • = 9x² - 2x³ + 2
    • What can you say about what f(x) is? How about g(x)?
      • f(x) = x³ + C
      • g(x) = 1/4 x⁴ - x + C
      • Antiderivatives aka indefinite integrals
  • Position (s), velocity (v), and acceleration (a)
    • v = ds/dt
    • a = dv/dt
    • Constant acceleration a
    • v = at + C
    • s = 1/2 at² + Ct + K
    • Meteor
      • Height = 60 miles
      • Speed = 30000 mi/hr
      • g (=a) = 32 ft/sec²
        
      • Height = -1/2 (32 ft/sec²)t² - 30000 mi/hr t + 60 mi
• Next
  • Normals (perpendiculars) to curves
  • Product and quotient rules
  • Higher-order derivatives
    • “Products and Quotients” part of section 3.3
    • Top of page 122 - bottom of 124

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