The Chain Rule

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• Questions?
  • Problem set graphs w/ R problem?
    • Use “curve” function
  • Antiderivative of y = 3/x - 5sinx?
    • Fact: antiderivative of 3/x = 3 lnx
      • (You’ll never need to know this again in this course)
    • Focus: antiderivative of -5 sinx
      • Hmm… what about 5 cosx?
      • Check: d( 5 cosx )/dx
        • = 5 d( cosx )/dx
        • = 5 ( -sinx )
        • = -5 sinx
• The Chain Rule
  • Section 3.6
    • Good for derivatives of functions within functions
      • aka compositions of functions
      • e.g., f( g(x) )
    • If f(u) = y, g(x) = u
      • Then f′(g(x)) g′(x) = dy/du du/dx = (f°g)′ = dy/dx
    • To use the chain rule:
      1. Treat inner function as unit, differentiate outer around it
      2. Multiply by derivative of inner
  • Examples: find f′(x) if…
    • f(x) = √(1-x²)
      • u = (1-x²), i.e., inner
      • y = √u i.e., outer
      • f(x) = (1-x²)^1/2 = u^1/2
      • f′(x) = 1/2 (1-x²)^-1/2 (-2x)
      • = -x (1-x²)^-1/2
      • = -x / √(1-x²)
    • f(x) = sin( x² + 3x - 7 )
      • f′(x) = cos( x² + 3x - 7 ) (2x + 3)
    • f(x) = cos( 5x )
      • Outer = cos
      • 5x = inner
      • f′(x) = -sin(5x) 5 = -5 sin(5x)
    • f(x) = cos⁵x = (cos x)⁵
      • f′(x) = 5 cos⁴x (-sinx) = -5cos⁴x sinx
    • f(x) = cos( x⁵ )
      • Outer = cos
      • Inner = x⁵
      • f′(x) = -sin(x⁵) 5x⁴ = -5x⁴ sin(x⁵)
    • f(x) = 1 / ( x³ + 3√x )
      • Do this with and without the quotient rule
      • f(x) = ( x³ + 3√x )^-1
      • Outer = u^-1
      • Inner = x³ + 3√x
      • f′(x) = -( x³ + 3√x )^-2 (3x² + 3/(2√x))
      • = (-3x² - 3/(2√x)) ( x³ + 3√x )^-2
      • (Do quotient rule version for Friday)
• Next
  • More chain rule examples
  • Implicit differentiation

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