The first three sections of this chapter have given us some new reasons to be interested in calculating derivatives: They provide us important information about and interpretations of the graphs of functions, and they play an important role in root-finding via Newton's Method. So far, we know how to calculate derivatives of polynomial functions and of exponential functions. But that hardly accounts for all the functions of interest. In the remainder of the chapter, we concentrate on expanding our set of tools for computing derivatives. We start with a problem that shows the need for our first step in this development.

4.4.1 The Growth Rate of Energy Consumption

The population of the United States (in millions) from 1982 to 1990 is closely approximated by an exponential growth formula of the form

where time `t` is measured in years from 1982 (i.e., `t=0` in that year), `k=0.00917`, and `P_0=232`. That is, a population of `232` million in 1982 grew at a rate of just under `1`% per year for the next eight years. The per capita energy consumption over the same period (in millions of BTU's per person) is roughly approximated by a linear formula of the form

where `m=3.4` and `b=302`. Thus, the total energy use `T` is approximated by the product of these two functions,

Suppose we want to estimate the growth rate of energy consumption at some time in the 1980's. How can we calculate `dT text[/] dt`? We know how to differentiate the population and per-capita-energy functions separately, but how does that help us find the derivative of their product?

Note 1 – Source

Activity 1

1. Suppose `gtext[(]x text[)]=x, and `htext[(]x text[)]` also equals `x`. Let `ftext[(]x text[)]=gtext[(]x text[)]htext[(]x text[)]`; thus, `ftext[(]x text[)]=x^2`. What are the derivatives of `ftext[(]x text[)]`, `gtext[(]x text[)]`, and `htext[(]x text[)]`?

2. Suppose `gtext[(]x text[)]=x, and `htext[(]x text[)]=x^2`. Let `ftext[(]x text[)]=gtext[(]x text[)]htext[(]x text[)]`; thus `ftext[(]x text[)]=x^3`. What are the derivatives of `ftext[(]x text[)]`, `gtext[(]x text[)]`, and `htext[(]x text[)]`?

3. Suppose `gtext[(]x text[)]=e^(2x), and `htext[(]x text[)]=e^(3x)`. Let `ftext[(]x text[)]=gtext[(]x text[)]htext[(]x text[)]`. What is `ftext[(]x text[)]` as an explicit function of `x`? What are the derivatives of `ftext[(]x text[)]`, `gtext[(]x text[)]`, and `htext[(]x text[)]`?

4. Try to give a formula describing `f'text[(]x text[)]` in terms of `g'text[(]x text[)]` and `h'text[(]x text[)]` that is valid for each of parts (a), (b), and (c).

Comment on Activity 1

Image credits