Differentiability of Composite Functions |
Compositefunctions are so common that we usually don't think to think to label them ascomposite functions. However, they arise any time a change in one quantityproduces a change in another which, in turn, produces a change in a thirdquantity. Does that sound confusing? Don't worry, an example will make thingsclear. An Example - For this example, we'll assume that the number of humansliving on the coast affects the number of whales in nearby coastal waters.Since whales eat plankton, the number of whales will affect the number ofplankton in the waters. Let'sbe more specific. Since whales don't like all the noise that people make, theymove out of an area when too many people move in. If we denote the number ofthousands of people by x and the number of whales by y, a simple model would bethat. Nowsince the whales are eating the plankton, more whales mean less plankton. If wemeasure the amount of plankton by z, then a simple model is that. Nowthe end result is that the number of people influence the number of whaleswhich influences the number of plankton. To see how the number of planktondepend on the number of people, we can compute that More generally, if we havetwo functions and , we call the newfunction the composite of and and denote it by .