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: