A power function has a variable x in the base and a constant for the power. An exponential function has a constant for the base and a variable for the power:
f(x) = ax.
In order to make life easier (we do that sometimes) we assume a is not 0, 1, or negative. If a is 0 then our function is f(x) = 0x, which is undefined when x = 0 and is 0 everywhere else. Not too interesting.
If a is 1 then our function is f(x) = 1x = 1, which is a constant function, so also not too interesting. If a is negative, the function f(x) = ax is too weird to deal with: f will be negative some places, positive some places, and undefined at a lot of places (such as when x = 0.5.).
The following are exponential functions:
f(x) = 2x
f(x) = ex
f(x) = (0.5)x
f(x) = xx
is not an exponential function because it has a variable for the base and the power.
When we think of "exponential function," a good default function to think of is
f(x) = ex.
If we look at the estimates from the previous exercise, we're estimating that when f(x) = ex,
It turns out this is accurate. The derivative of the function f(x) = ex is
f'(x) = ex.
f(x) = ex is its own derivative.
If f(x) = ex then f ' (4) = e4.
Of course, there are exponential functions with other bases besides e. We'll give the derivative rule here, and the reasons after we talk about the chain rule.
If f(x) = ax, then f ' (x) = ax ln a.
If f(x) = 2x, then f ' (x) = 2x ln 2. To find the value of the derivative at a specific value of x, we plug that value in for x in the derivative function:
f ' (4) = 24 ln 2 = 16ln2.