The textbook I am using to teach Calculus I includes in the exercises of most chapters a number of interesting real-world applications of the concepts from that chapter. However, the chapter on the derivative of the natural logarithm is remarkably abstract in its exercises.
Are there not scenarios in which it would be useful to differentiate a logarithm to answer a real-world problem? Something to do with determining the stimuli needed to accomplish a particular exponential rate of growth?
Or is differentiating natural logarithms primarily motivated by its usefulness for simplifying differentiation using logarithmic differentiation?
Since a number of practical exponential growth and logarithm exercises problems revolve around population growth, I've been trying to contrive an example in those terms. But it feels very odd and abstract:
Exponential function: What will be the population after x years? Derivative of exponential: How quickly will the population be growing x years from now? Logarithmic function: How many years will it take to reach a particular target population? Derivative of logarithm: How much would increasing or decreasing a given target population affect the time it takes to reach it?
Is that even really a practical question? How do I make it interesting?
I've tried searching this question here and on Google, but haven't found anything. Thanks in advance for your help!