Exponential functions have the general form:
with two constants a and b (called base). It’s quite common to use Euler’s number e = 2.7182… as the base and the exponential function expressed as such:
with two constants a and c. Converting from one form to the other is not that difficult, just use ec = b or c = ln(b). Here’s how it works:
As for the plot, you should keep two special cases in mind. For b > 1 (which corresponds to c > 0 in case of base e), the function goes through the point P(0,a) and goes to infinity as x goes to infinity.
(Exponential function with b > 1 or c > 0. For example: f(x) = 8·3x)
This is exponential growth. When 0 < b < 1 (or c < 0) this turns into exponential decline. The function again goes through the point P(0,a), but approaches zero as x goes to infinity.
(Exponential function with 0 < b < 1 or c < 0. For example: f(x) = 0.5x)
Here’s how the differentiation of exponential functions works. Given the function:
The first derivative is:
For the case of base e:
You should remember both formulas. Note that the exponential functions have the unique property that their first derivative (slope) is proportional to the function value (height above x-axis). So the higher the curve, the sharper it rises. This is why exponential growth is so explosive.
If exponential functions are combined with power or polynomial functions, just use the sum rule.
(This was an excerpt from the FREE ebook “Math Shorts – Derivatives”)