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# Class Note for MATH 425A with Professor Laetsch at UA

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This 1 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Arizona taught by a professor in Fall. Since its upload, it has received 31 views.

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Date Created: 02/06/15
Math 4255251 Derivative of Inverse Function 2008 Refer to Theorem 4 11 on the derivative of the inverse functions The following list contains the assumptions that are either explicitly made in this theorem or which are immediate consequences of the assumptions made 1 f is a continuous function defined on an interval I Q R 2 f is onetoone so it has an inverse f391 fI gt R For convenience we will denote the inverse function by F 3 a is an interior point of I 4 f is differentiable at a and f 39a 0 5 f a is an interior point of f I and F is continuous at f a We will write b f a We want to prove that F the inverse function is differentiable at b and find the derivative To that end we consider the difference quotient for F Let y be a point in a neighborhood of b with y i b and let x Fy so that y fx Then Fy Fb 1 1 1 yb yb fxfa fFyfa FyFb xa Fya 1 Show that the equations above make sense by explaining why F y F b i 0 2 Show that the limit of the expression on the right exists and determine what it is using the assumptions stated above along with Theorem 336 and especially Theorem 337 on the limit of the composition of functions Do this in careful detail in particular define explicitly the functions you are using here which play the role of the functions f and g in Theorem 337 and state what numbers play the role of A and B 3 Prove that F is differentiable at b and nd the derivative

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