05 2conditions

Conditions of Differentiability

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The derivative is only defined if the function is continuous and smooth.

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This means the derivative does not exist (is not defined) in the following conditions:

  • where there is a vertical asymptote
  • where there is a break or a hole
  • where there is a sharp corner (cusp)
  • at the endpoints of the domain
05.2condition1.gif
  • $f'(x)$ is undefined at $x = 1$ and at $x = 3$
    • Because of the cusps ("pointy bits"), f(x) is not smooth at those two points

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05.2condition2.gif
  • $g'(x)$ is undefined at $x = 1$
    • Because of the endpoint
  • $g'(x)$ is undefined for $x < 1$
    • Because g(x) is undefined for those values
  • $g'(x)$ is undefined at $x = 3.5$
    • Because of the hole (discontinuity)

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05.2condition3.gif
  • $h'(x)$ is undefined at $x = 2$
    • Because h(x) is undefined at the asymptote

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