The Chain Rule and Implicit Differentiation

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• Questions?
• The Chain Rule
  • Examples: find f′(x) if…
    • f(x) = 1 / ( x³ + 3√x ) = ( x³ + 3√x )^-1
      • Chain rule yields f′(x) = (-3x² - 3/(2√x)) ( x³ + 3√x )^-2
      • Quotient rule agrees
    • f(x) = √( sin(x²+1) + cos(x²-1) )
      • Chain rule. Eventually needs derivatives of inner sin and cos expressions, so...
        • sin(x²+1): derivative of outer times derivative of inner = cos(x²+1) 2x
        • cos(x²-1): derivative = -sin(x²-1) 2x
      • Now differentiate √(....)
        • Derivative = 1/2 ( sin(x²+1) + cos(x²-1) )^-1/2 (cos(x²+1) 2x - sin(x²-1) 2x)
        • = ( sin(x²+1) + cos(x²-1) )^-1/2 (x cos(x²+1) - x sin(x²-1))
        • = ( sin(x²+1) + cos(x²-1) )^-1/2 x (cos(x²+1) - sin(x²-1))
  • Side note: notation sinx is not the same as using juxtaposition for multiplication. Notation for trig and logarithm functions is function application, even though it’s a little different than the notation for applying other functions.
    • f(x) vs sinx
    • e.g., x²sinx sinx³ ≠ sinx⁵
  • Economics example
    • Suppose the quantity of some product that people are willing to buy is inversely related to its price:
      • q = 1/p
    • What is dq/dp, and what does it reflect?
      • Sensitivity of people to price
        
    • If the product is some new invention, its price is likely to drop sharply over time, maybe
      • p = 1 - (t-1)^1/3
      • Still have q = 1/p
    • What is dq/dt, and what does it have to do with composition of functions and the chain rule?
      • q(t) = q(p(t)) = 1 / (1 - (t-1)^1/3) = (1 - (t-1)^1/3)^-1
      • dq/dt = -(1 - (t-1)^1/3)^-2 (-1/3 (t-1)^-2/3 )
        
• Implicit Differentiation
  • Section 3.7
  • Consider the circle defined by x² + y² = 1
    
    • Does it have a derivative? If so, what is it?
    • d/dx(x² + y² ) = d/dx(1)
    • 2x + 2y dy/dx = 0
    • 2y dy/dx = -2x
    • dy/dx = -x/y
  • Equations of the form ax² + by² + cxy + dx + ey + f = 0
    • 2D version of 3D “quadric surfaces” in computer graphics, which can describe spheres, cylinders, 3D generalizations of other conic sections, etc.
    • General form for all conic sections including rotations and degenerate cases
    • For example 3x² - y² - 9 = 0
      • dy/dx = ?
      • 6x - 2y dy/dx = 0
      • 3x - y dy/dx = 0
      • 3x = y dy/dx
      • 3x/y = dy/dx
• Next
  • Related rates problems
  • Section 3.8

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