# L10 - Lecture 10 The Derivative Function In the last...

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Lecture 10 — The Derivative Function In the last examples of Lecture 9, we saw that, from a function f , we could derive a new function that represents the slope (or instantaneous rate of change) of f at each x = a . This function is called the derivative of f , and we give it the special nota- tion f 0 . Thus, f 0 ( a ) = or equivalently, if we let h = x - a , f 0 ( a ) = Other notations Interpretations

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The process of ﬁnding the derivative of a function is called differentiation . Note the derivative is deﬁned as a limit, which may or may not exist. If f 0 ( a ) exists, we say that f is differentiable at a . We say it is differentiable on an interval I if it is differentiable at each point in I . On what interval(s) is the function V ( t ) = ( t 2 + t ) 1 3 differentiable?
Find the values of the constants a,c that make the following function differentiable: g ( x ) = ± x 2 x 1 a + cx x > 1 Determine whether the function is differentiable at t = 0 : E ( t ) = ( te - 1 t 2 t 6 = 0 0 t = 0

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Theorem: If a function f is differentiable at a , then it is continuous at a . Beyond calculus: The world is full of wild functions
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## This note was uploaded on 05/27/2011 for the course MAC 2311 taught by Professor All during the Spring '08 term at University of Florida.

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L10 - Lecture 10 The Derivative Function In the last...

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