Poop and More Poop Essay

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The Derivative of a Composite Function Definition : If f and g are two functions defined by y = f(u) and u = g(x) respectively then a function defined by y = f [g(x)] or fog(x) is called a composite function or a function of a function. Corollary : If y = f (u), u = g (v) and v = h (x) where f, g and h are differentiable functions of u, v and x respectively, then Show that = | | Theorem : If f and g are differentiable and are defined by y = f (u) and u = g (x), then the composite function y = f [ g (x) ] is differentiable and we have The Chain Rule | As a motivation for the chain rule, consider the function f(x) = (1+x2)10. Since f(x) is a polynomial function, we know from previous pages that f'(x) exists. Naturally one may ask for an explicit formula for it. One tedious way to do this is to develop (1+x2)10 using the Binomial Formula and then take the derivative. Of course, it is possible to do this, but it won't be much fun. But what if we have to deal with (1+x2)100! Then I hope you agree that the Binomial Formula is not the way to go anymore. So what do we do? The answer is given by the Chain Rule. Before we discuss the Chain Rule formula, let us give another example. Example. Let us find the derivative of . One way to do that is through some trigonometric identities. Indeed, we have So we will use the product formula to get which implies Using the trigonometric formula , we get Once this is done, you may ask about the derivative of ? The answer can be found using similar trigonometric identities, but the calculations are not as easy as before. Again we will see how the Chain Rule formula will answer this question in an elegant way. In both examples, the function f(x) may be viewed as: where g(x) = 1+x2 and h(x) = x10 in the first example, and and g(x) = 2x in the second. We say that f(x) is the composition

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