The title of this chapter might well be the title of our whole first course in calculus. Here we will consolidate and build on what we have accomplished already, laying the ground work for the rest of the first course.

By addressing problems that appeared naturally as differential equations, we found a need for addressing the (easier) inverse problem of calculating derivatives of known functions. The problems in this chapter lead directly to the need for calculating derivatives — in many cases, derivatives of functions we have not yet differentiated. As we find out how to differentiate these new functions, we will expand our repertoire of derivative formulas and thereby enhance our ability to solve differential equations as well.

We begin by studying what derivatives tell us about the graphs of functions and what graphs tell us about derivatives.  We will find an intimate connection with problems of optimization, that is, of finding the best (or worst) way to do something. That connection leads to the problem of solving nonlinear algebraic equations, not a difficult task with a computer or a graphing calculator. Indeed, when we ask what Solve function might be doing (in a computer algebra system or on a calculator), we find an elegant and practical application of the derivative called Newton's Method.

Throughout the chapter we study prototypical applications — manufacturing costs, growth of energy consumption, physical properties of light. In each case we find that our mathematical models require us to expand our repertoire of derivative formulas. By the end of the chapter, our growing list of formulas will include the Product Rule, the General Power Rule, and the most important derivative formula of all: the Chain Rule.