Section 16

Guide to differentiate and analyze

Use cues in the function to decide which technique to try first.

View the full section in the index →

Signal → method

  1. Signal

    Polynomial or power xnx^n

    Method

    Power rule

  2. Signal

    Product of functions

    Method

    Product rule

  3. Signal

    Quotient

    Method

    Quotient rule

  4. Signal

    Composition f(g(x))f(g(x))

    Method

    Chain rule

  5. Signal

    fgf^g with variable exponent

    Method

    Logarithmic differentiation

  6. Signal

    Implicit equation F(x,y)=0F(x,y)=0

    Method

    Implicit differentiation

  7. Signal

    Limit 0/00/0 or ∞/∞\infty/\infty

    Method

    L'Hôpital or algebra

  8. Signal

    Optimize on [a,b][a,b]

    Method

    Critical points + interval endpoints

  9. Signal

    Approximate near aa

    Method

    Linearization or Taylor

Checklist

  1. Domain and symmetries.
  2. Axis intercepts.
  3. Vertical, horizontal and oblique asymptotes.
  4. f′f': intervals of increase/decrease and critical points.
  5. f′′f'': concavity and inflection points.
  6. Sign chart and final sketch.

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