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Derivatives of Exponentials

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.

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Derivatives of Exponentials

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  1. Derivatives of Exponentials Rates of Change for Quantities that Grow or Decrease in an Exponential Way Dr. E. Fuller Dept of Mathematics WVU

  2. 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£L£b

  3. Change in ex • So how does change occur in exponential cases? • In class we saw that if f(x) = ex then for every x • So we can approximate change for values of x by approximating this expression

  4. 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? (eh-1)/h as before

  5. The Derivative of ex • If f(x) = ex, then f 0(x) = ex • So the value of the slope at any point is equal to the value of the function • ex is the only function for which this is true

  6. 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) = e2x f 0(x) ¼ 2f(x)

  7. A General Formula • In general, for f(x) = ekx where k is a constant, f0(x) = kekx. • In other words, for f(x) = ekx, 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

  8. 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) = Aekx 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

  9. The Chain Rule for eu(x) • The most general rule for exponentials says that if f(x) = eu(x), then f 0(x) = eu(x)u0(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

  10. An Example • Ex: If f(x) = ex2, find f 0(2). • Solution: f0(x) = ex2(2x) = 2xex2 and so f 0(2) = 2(2)e22=4e4

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