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# 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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