Outline

1. introduction
   1. average rate of change over an interval
   2. instantaneous rate of change
      1. at a point
      2. as a function of x
   3. difference quotients
      1. at a point
      2. as a function of x (and h)
   4. marginals (small h approximation to the limit)
   5. introduction to the "operator" concept
2. limits and the definition of the derivative
   1. limits at infinity -- asymptotes
   2. limits at a point
      1. limits from the left and the right (aka below and above)
      2. two-sided limits
   3. limit laws
      1. limit of a constant
      2. limit of a scalar multiple
      3. limits of sums and linear combinations
      4. limits of products and quotients
   4. limits of difference quotients
      1. at points
      2. at variable x
   5. the formal definition of the derivative
      1. def
      2. notations
         1. Newton's dots --> primes
         2. Leibniz' notation
         3. the modern D
      3. using the definition to find derivatives
         1. power functions
         2. sine and cosine
         3. exp(x)
      4. linearity of the D operator
3. differentiation rules and applications
   1. derivatives of constants and linear functions
   2. the power rule
   3. derivatives of linear combinations
   4. the product rule
   5. the chain rule
   6. the quotient rule
4. techniques
   1. implicit differentiation
   2. related rates
   3. derivatives of inverse functions
5. applications -- theory and practice
   1. intermediate value theorem
   2. optimization