Exponential Derivatives Discussion

This discussion explores derivatives of exponential functions, building on ideas in “Derivative of the Exponential Function” in section 3.9 of our textbook. This reading’s main result is a formula for the derivative of e^x, namely

This discussion looks at a few applications and extensions of this formula.

Please post at least one response to at least one of the following by class time on Friday, October 9.

A General Exponential Derivative

There are lots of exponential functions besides e^x, for example 2^x, 3^x, 1.947^x, etc., and they all have derivatives. Can you find a general formula for the derivative of a^x, where a could be any constant?

To start the discussion, remember that a = e^{ln a}, where “ln” is the natural logarithm; since a is a constant, so is ln a. Can you use this to rewrite a^x? If so, can you start to see how to differentiate your rewritten form?

Antiderivatives

Can you “run the rule backwards” to find an antiderivative for e^x?

Can you do a similar thing for a general exponential function a^x, where a could be any constant? (Having ideas in the “General Exponential Derivative” part above will probably help with this.)

What about an antiderivative for 3e^x? (I’ll admit that I’m putting this in the discussion just because it’s an excuse to identify a “constant multiple” antiderivative rule, which is a good thing to know in general, not because there’s something particularly important about constant multiples of exponential functions.)

Population Growth

Many biological populations grow approximately exponentially, as long as they have enough food available, don’t have predators, and don’t face other constraints (these are big “as long as” assumptions, of course). Examples include the initial phase of growth of bacteria in a culture, yeast in brewing beer or wine, and (likely of current interest) infections in a pandemic. Usually mathematicians start with a model of how the population changes (i.e., an assumption about the derivative of the population) and deduce an exponential function for the size of the population from there. We don’t quite know enough calculus to do this yet (although we might by the end of this course), but we can go the other way: start by assuming an exponential population size, and infer something about how the population changes from that.

For example, the number of bacteria in a culture flask t hours after starting the culture might be given by

What is the rate of change in population as a function of time?

More interesting, what is the ratio of the rate of change to the population (i.e., P′(t)/P(t))?

Can you generalize the above example, i.e., if the equation is

where A and c could be any constants, can you find the ratio of rate of change to population?

Can you interpret the ratio as a “model” of how the organisms reproduce? This is probably a good discussion question, i.e., a place where lots of people can chime in with different ideas, comments on ideas, questions about ideas, etc.