Derivatives of Exponentials Rates of Change for Quantities that Grow or Decrease in an Exponential Way Dr. E. Fuller Dept of Mathematics WVU
Exponential Growth/Decrease Recall that a quantity is “exponential” if its value increases or decreases as a product of a fixed base multiplied by itself some number of times Value = (initial amount)b £ b £ b £ £ b
Change in e x So how does change occur in exponential cases? In class we saw that if f(x) = e x then for every x So we can approximate change for values of x by approximating this expression
Approximating Change Listed below are approximations of f 0 (x) for x between –3 and 3 (h=.0001) What do you notice about this graph? (e h -1)/h as before
The Derivative of e x If f(x) = e x, then f 0 (x) = e x So the value of the slope at any point is equal to the value of the function e x is the only function for which this is true
Proportional Change What this means is that the change in f is a constant multiple of the value of f We can change the exponent slightly to see this effect Ex: f(x) = e 2x f 0 (x) ¼ 2f(x)
A General Formula In general, for f(x) = e kx where k is a constant, f 0 (x) = ke kx. In other words, for f(x) = e kx, f 0 = kf (the change in f is proportional to the value of f) Ex: f(x) = e.25x then f 0 (x)=.25e.25x Ex: f(x) = e -2x, then f 0 (x) = -2e -2x
A General Fact More surprising, it works the other way as well: If f 0 (x) = kf(x) for some constant k, then f(x) = Ae kx for some constant A. Note that A = f(0) (the initial value of f) So if you know a quantity is changing at a rate proportional to its value, that quantity must be exponential
The Chain Rule for e u(x) The most general rule for exponentials says that if f(x) = e u(x), then f 0 (x) = e u(x) u 0 (x) This is the Chain Rule for exponential derivatives. All the previous examples are special cases of this. To test your understanding, work the Exercises Exercises
An Example Ex: If f(x) = e x 2, find f 0 (2). Solution: f 0 (x) = e x 2 (2x) = 2xe x 2 and so f 0 (2) = 2(2)e 2 2 =4e 4