The slope of a line is defined to be the rise divided by the run. For a curved line, however, the slope has to be defined at a point. Lines (curved or straight, but not infinitely steep) can always be thought of as functions of a single variable. We call the slope of a line evaluated at any given point its derivative, and call the process of finding that slope taking the derivative of the function.
Later we'll say a few words about multivariate (vector) differential calculus, but that is mostly beyond the scope of this course.
The definition of the derivative of a function is:
First, note that:
Second, differentiation is linear. That is:
Third, suppose that
(the product of two functions). Then
We can easily and directly compute the derivative of a mononomial:
Again it is beyond the scope of this short review to completely
rederive all of the results of a calculus class, but from what has been
presented already one can see how one can systematically proceed. We
conclude, therefore, with a simple table of useful derivatives and
results in summary (including those above):