SECTION 3.2: Exponential and Logarithmic Functions

The Exponential Function:

The graph of an exponential function increases slowly for x<0>0. This implies that the derivative of a function f(x) = a^x, is small for x<0>0. Since the function is increasing for all values of x, the graph of the derivative always remains above the x-axis.

At a = 2.718... it is found that d/dx (a^x) = a^x. For this reason a is defined as the number e.
This implies that d/dx (e^x) = e^x.

The exponential rule of differentiation states that
d/dx (a^x) = (ln a)*(a^x)

Since ln a is a constant, it can be said that the derivative of the exponential function (a^x) is proportional to the original function.

The Derivative of Natural Logarithm:

The graph of f(x) = ln x, shows that ln x is always increasing. Therefore its derivative is positive. The graph of f(x) is concave down, therefore its derivative must be decreasing.

ln x is very large near x=0 and very small at x = infinity.

The natural logarithm rule of differentiation states that,
d/dx (ln x) = 1/x

The difficult part about this section was understanding exactly why those derivative formulae are obtained. Beyond seeing the graphical explanations, there isn't enough to really understand how ones arrives at those formulae.

Logarithms are another method used to interpret graphical trends and hence can be useful in Economics.