Answers and Explanations
which is times a number (the limit of as h goes to 0) which is a constant independent of x, depending only on the base a of the exponential.
Therefore, the derivative of an exponential function is just a constant times the function value. What the constant is depends on what the base of the exponential function is.
The number e is that value of the base which yields the constant 1, so that the derivative of the function is actually equal to itself.
Moreover, the derivative of the more general function is . If we were using some other number a as the base instead of e, there would be an additional constant out front.
Not only does this make it more natural and convenient to use e as the base in exponentials and logarithms, but it relates back to the compound interest interpretation given earlier. If a bank account is growing under compound interest, with an interest rate of R per unit time, that means that at any instant in time the growth rate is R times the current balance. We've already seen that the balance at the end of the period is where B is the beginning balance. More generally, to figure out the balance somewhere in the middle of the time period, let t denote the fraction of the time period that has passed; it turns out that the balance at that time is . The derivative of this function is which is R times the function value; this is just another way of saying that the growth rate is R times the current balance.
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