Lect13 - The Derivative as a Function Let's extend the...

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92.131 Lecture 13 1 of 14 Ronald Brent © 2008 All rights reserved. The Derivative as a Function Let’s extend the definition of the derivative at a point to a derivative function. h x f h x f x f h ) ( ) ( lim ) ( 0 + = Notes: In essence we are simply replacing a with x creating a formula for the rate function associated with ) ( x f . We can’t just let 0 = h , some algebraic manipulation or trick is needed. It may be impossible to find the derivative, especially if ) ( x f has a very complicated formula. In this case a numerical method is needed. The domain of ) ( x f is the set of inputs x for which the limit exists. In many cases this is the full domain of ) ( x f . However in some cases it isn’t, for example the function x x f = ) ( has no derivative at 0 = x .
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92.131 Lecture 13 2 of 14 Ronald Brent © 2008 All rights reserved. Example: Let 2 ) ( x x f = using the limit definition of the derivative show x x f 2 ) ( = h x f h x f x f h ) ( ) ( lim ) ( 0 + = Since 2 2 2 2 ) ( ) ( h h x x h x h x f + + = + = + , x h x h h xh h x h xh x h x f h x f h h h h 2 ) 2 ( lim 2 lim 2 lim ) ( ) ( lim 0 2 0 2 2 2 0 0 = + = + = + + = + x 2 ) ( x x f = x x f 2 ) ( = 0 0 0 ± 1 1 ± 2 ± 2 4 ± 4 ± 3 9 ± 6
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92.131 Lecture 13 3 of 14 Ronald Brent © 2008 All rights reserved.
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This note was uploaded on 02/13/2012 for the course MATH 92.131 taught by Professor Staff during the Fall '09 term at UMass Lowell.

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Lect13 - The Derivative as a Function Let's extend the...

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