Now that we understand the notion of a derivative, we hope to actually compute it for
functions built out of the elementary
functions, including the linear, polynomial,
rational, power, and trigonometric functions.
We begin by calculating the derivatives of the elementary
functions. Then
we introduce several important techniques of differentiation, including the
product rule, the quotient rule, the chain rule, and implicit differentiation.
These are the techniques that allow us to compute the derivative of a complicated
function in terms of the derivatives of the elementary functions that make it up.
Recall that we originally understood a derivative as the slope at a particular point on the
graph of a function. Armed with the tools in this SparkNote, we will be able to compute
many derivatives without any reference to their graphs. Thus we will be able to deduce
geometric information from symbolic manipulation. The fact that this works is a glimpse
of what makes calculus so useful and beautiful.
