3.2: Exponential and Logarithmic Functions.

Main Points
The derivative graph of an exponential function looks exactly like the original graph. For any function in the form a^x, their derivative graphs are proportional to the original, but for the function e^x the derivative graph is exactly the same. Thus, the exponential rule is d/dx(a^x) = (ln a)a^x. Finally why is e is useful base is explained: the fact that the constant of proportionality is 1 when a = e. The derivative of ln x is 1/x.

Challenges
Why is the derivative at the point x=0 of the 2^x function approx. 0.693 rather than 0? How exactly are functions where x is the power differentiated?

Reflections
With the examples that we have been doing in class it is becoming more and more obvious how differentiation fits seamlessly into everyday life though it may of course be very theoretical with hypothesized figures. I never really thought differentiation was what was used to compute the demand and supply functions and thus interpretations in economics.