Numerical Methods in Computational Finance. Daniel J. Duffy

(1.7)

      This limit may not exist at certain points, and it is possible to define right-hand and left-hand limits, that is, one-sided derivatives.

      Some results that we learn in high school are:

      (1.8)

A composite function is a function that we can differentiate using the chain rule that we state as follows:

      (1.9)

      A simple example of use is:

      More challenging examples of composite functions are:

      1.3.1 Taylor's Theorem

      Taylor's theorem allows us to expand a function as a series involving higher-order derivatives of a function. We take the Cauchy form (with exact remainder):

       

      (1.10)

      and:

We conclude with a discussion of the exponential function. It is the only function that is the same as its derivative. To see this, we use the formal definition (1.7) of a derivative (and noting that

):

      We summarise some useful properties of the exponential function: