The operation that takes a function f and returns a new function f' representing the rate of change of the old function is a wonderful mathematical tool known as the derivative. This concept is also relevant in biology: iff'(t) represents the number of bacteria in a culture dish at time t, then f '(t) is the rate at which the bacteria reproduce themselves. As long as there is enough space and resources in the dish, then (for bacteria) f '(t) will be approximately proportional to f '(t). In fact, this determines which function f '(t) is required to be, up to a constant!
Before continuing, it is important to give a few properties of our wonderful mathematical tool. For n > 2, this calculation may be performed by the binomial theorem, a general result that expresses (x + y)n in terms of powers of x and y. This can even be made to work for fractional n, although more care is necessary. One finds for all n the result
For n = 2, this reproduces the result of the first calculation from this chapter. This is called the power rule.
A crucial property of the derivative is linearity, which means that where f,h are any two functions whose derivatives exist, and c is any constant. There are also convenient rules for calculating derivatives of more complicated functions, if you know how to do it for simple functions. We mention the product rule, the rule for finding the derivative of a product function:
and the chain rule, the rule for finding the derivative of a composed function:
You might say that Equation 2.7 looks great for calculating the derivative of a product; but what about quotients? What is (f/g)'? To answer this, note that where g-1 = 1/g by definition (the -1 exponent does not mean the inverse function).
We could apply the product rule if only we knew -^-(g-1) •
The latter, however, is calculated by the chain rule and the power rule, together! Let's do this explicitly, to solidify our understanding.
The power rule (Equation 2.6) for n = -1 gives us = •
Then Equation 2.8 tells us that
The product rule now gives
This formula is called the quotient rule. It follows from the other rules.
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