Math 19: Calculus Winter 2008 Instructor: Jennifer KlokeLecture Outlines (The Chain Rule)Monday, February 11Announcements1. Homework 4 due on Wednesday, 2/13/08.RecapLast time we learned how to take the derivative of trigonometric functions. Recall:1.ddxsin(x) = cos(x).2.ddxcos(x) = −sin(x).3.ddxtan(x) = sec2(x).Composition of FunctionsSo far, you know how to differentiate powers of x, exponential functions, trigonometric func-tions, and sums, products, and quotients of those functions. But what about sin(x2), ecos(x),or√1 + ex? Specifically, what about functions that are the composition of these simplierfunctions?Recall f rom the first week of class that we learned how to take a more complecatedfunction and break it down into a composition of simplier functions using the ◦ symbol.For instance, the function sin(x2) can be rewritten as (f ◦ g)(x) where f (x) = sin(x) andg(x) = x2.ExamplesFor each of the following functions, write it as a composition (f ◦ g)(x) for some functionsf(x) and g(x).1. sin(p(x))2.√x2+ 13. e2 cos(x)1The Chain RuleThe Chain Rule tells us how to differentiate a composition of differentiable functions:ddx(f ◦ g)(x) =ddxf(g(x)) = f0(g(x))g0(x).The proof is in your textbook and I’m not going to cover that in class.The chain rule can be remembered as ”The derivative of the outside (taken at the inside)times the derivative of the inside.”This fact is amazingly useful but often causes problems for people who see this for thefirst time. Let’s just do lots of examples today to solidify your understanding of this concept.Examples1. Computeddxsin(p(x)).2. Computeddx√x2+ 1.3. Computeddxe2 cos(x).Examples of Putting everything together1. Find the derivative of xe−x2.2. Find the derivative of x2− 2 sin(3xex).3. (My way of computing derivatives of quotients:) Find the derivative ofx3+5x+1ex−1.Things to Know/MemorizeIf f(x) and g(x) are differentiable functions, then the chain rule gives us:ddx(f ◦ g)(x) =ddxf(g(x)) =
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