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.